Cross-multiplication

Common divisors method

Task. Find the values ​​of the expressions:

Task. Find the values ​​of the expressions:

To estimate how colossal gains the least common multiple method gives, try calculating the same examples using the criss-cross method.

Common denominator of fractions

Without a calculator, of course. I think after that comments will be superfluous.

See also:

I originally wanted to include methods for casting to common denominator in the paragraph "Addition and subtraction of fractions". But there was so much information, and its importance is so great (after all, common denominators are not only for numeric fractions) that it is better to study this issue separately.

So, let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The basic property of a fraction comes to the rescue, which, recall, sounds like this:

The fraction will not change if its numerator and denominator are multiplied by the same nonzero number.

Thus, if you choose the right factors, the denominators of the fractions will become equal - this process is called. And the required numbers, "leveling" the denominators, are called.

Why do you even need to bring fractions to a common denominator? Here are just a few reasons:

1. Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
2. Comparison of fractions. Sometimes converting to a common denominator makes this task much easier;
3. Solving problems for shares and percentages. Percentages are, in fact, common expressions that contain fractions.

There are many ways to find numbers that, when multiplied by, make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, efficiency.

Cross-multiplication

The simplest and reliable way which is guaranteed to flatten the denominators. We will go ahead: we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators. Take a look:

Task. Find the values ​​of the expressions:

Consider the denominators of neighboring fractions as additional factors. We get:

Yes, it's that simple. If you are just starting to learn fractions, it is better to work with this particular method - this way you will insure yourself against many mistakes and are guaranteed to get the result.

The only drawback of this method is that you have to count a lot, because the denominators are multiplied "ahead of time", and as a result, very large numbers can be obtained. This is the price to pay for reliability.

Common divisors method

This technique helps to greatly reduce the calculations, but, unfortunately, it is rarely used. The method is as follows:

1. Before you go ahead (that is, the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided by the other.
2. The number obtained as a result of such division will be an additional factor for the fraction with a lower denominator.
3. In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is the savings. At the same time, the probability of error is sharply reduced.

Task. Find the values ​​of the expressions:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is evenly divisible by the other, we apply the common factors method. We have:

Note that the second fraction was never multiplied by anything at all. In fact, we have cut the amount of computation in half!

By the way, I took the fractions in this example for a reason. If you're curious, try counting them crosswise. After the reduction, the answers will be the same, but there will be much more work.

This is the strength of the method of common divisors, but, I repeat, it can be applied only when one of the denominators is divisible by the other without a remainder. Which is rare enough.

Least Common Multiple Method

When we bring fractions to a common denominator, we are essentially trying to find a number that is divisible by each of the denominators. Then we bring the denominators of both fractions to this number.

There are a lot of such numbers, and the smallest of them will not necessarily be equal to the direct product of the denominators of the original fractions, as it is assumed in the "criss-cross" method.

For example, for the denominators 8 and 12, the number 24 is quite suitable, since 24: 8 = 3; 24: 12 = 2. This number is much less than the product 8 · 12 = 96.

The smallest number that is divisible by each of the denominators is called their (LCM).

Notation: the least common multiple of a and b is denoted by LCM (a; b). For example, LCM (16; 24) = 48; LCM (8; 12) = 24.

If you can find such a number, the total amount of computation will be minimal. Take a look at examples:

How to find the lowest common denominator

Find the values ​​of the expressions:

Note that 234 = 117 · 2; 351 = 117 · 3. The factors 2 and 3 are coprime (they have no common divisors except 1), and the factor 117 is common. Therefore, the LCM (234; 351) = 117 · 2 · 3 = 702.

Similarly, 15 = 5 · 3; 20 = 5 · 4. The factors 3 and 4 are relatively prime, and the factor 5 is common. Therefore, LCM (15; 20) = 5 3 4 = 60.

Now we bring the fractions to common denominators:

Note how helpful factoring the original denominators was:

1. Having found the same factors, we immediately arrived at the least common multiple, which, generally speaking, is a nontrivial problem;
2. From the resulting expansion, you can find out which factors are "missing" for each of the fractions. For example, 234 3 = 702, therefore, for the first fraction, the additional factor is 3.

Do not think that such complex fractions will not be in the real examples. They meet all the time, and the above tasks are not the limit!

The only problem is how to find this very NOC. Sometimes everything is found in a few seconds, literally "by eye", but on the whole this is a complex computational problem that requires separate consideration. We will not touch on this here.

See also:

Common denominator of fractions

Initially, I wanted to include common denominator methods in the Adding and Subtracting Fractions paragraph. But there was so much information, and its importance is so great (after all, common denominators are not only for numeric fractions) that it is better to study this issue separately.

So, let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The basic property of a fraction comes to the rescue, which, recall, sounds like this:

The fraction will not change if its numerator and denominator are multiplied by the same nonzero number.

Thus, if you choose the right factors, the denominators of the fractions will become equal - this process is called. And the required numbers, "leveling" the denominators, are called.

Why do you even need to bring fractions to a common denominator?

Common denominator, concept and definition.

Here are just a few reasons:

1. Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
2. Comparison of fractions. Sometimes converting to a common denominator makes this task much easier;
3. Solving problems for shares and percentages. Percentages are, in fact, common expressions that contain fractions.

There are many ways to find numbers that, when multiplied by, make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, efficiency.

Cross-multiplication

The easiest and most reliable way that is guaranteed to align the denominators. We will go ahead: we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators. Take a look:

Task. Find the values ​​of the expressions:

Consider the denominators of neighboring fractions as additional factors. We get:

Yes, it's that simple. If you are just starting to learn fractions, it is better to work with this particular method - this way you will insure yourself against many mistakes and are guaranteed to get the result.

The only drawback of this method is that you have to count a lot, because the denominators are multiplied "ahead of time", and as a result, very large numbers can be obtained. This is the price to pay for reliability.

Common divisors method

This technique helps to greatly reduce the calculations, but, unfortunately, it is rarely used. The method is as follows:

1. Before you go ahead (that is, the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided by the other.
2. The number obtained as a result of such division will be an additional factor for the fraction with a lower denominator.
3. In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is the savings. At the same time, the probability of error is sharply reduced.

Task. Find the values ​​of the expressions:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is evenly divisible by the other, we apply the common factors method. We have:

Note that the second fraction was never multiplied by anything at all. In fact, we have cut the amount of computation in half!

By the way, I took the fractions in this example for a reason. If you're curious, try counting them crosswise. After the reduction, the answers will be the same, but there will be much more work.

This is the strength of the method of common divisors, but, I repeat, it can be applied only when one of the denominators is divisible by the other without a remainder. Which is rare enough.

Least Common Multiple Method

When we bring fractions to a common denominator, we are essentially trying to find a number that is divisible by each of the denominators. Then we bring the denominators of both fractions to this number.

There are a lot of such numbers, and the smallest of them will not necessarily be equal to the direct product of the denominators of the original fractions, as it is assumed in the "criss-cross" method.

For example, for the denominators 8 and 12, the number 24 is quite suitable, since 24: 8 = 3; 24: 12 = 2. This number is much less than the product 8 · 12 = 96.

The smallest number that is divisible by each of the denominators is called their (LCM).

Notation: the least common multiple of a and b is denoted by LCM (a; b). For example, LCM (16; 24) = 48; LCM (8; 12) = 24.

If you can find such a number, the total amount of computation will be minimal. Take a look at examples:

Task. Find the values ​​of the expressions:

Note that 234 = 117 · 2; 351 = 117 · 3. The factors 2 and 3 are coprime (they have no common divisors except 1), and the factor 117 is common. Therefore, the LCM (234; 351) = 117 · 2 · 3 = 702.

Similarly, 15 = 5 · 3; 20 = 5 · 4. The factors 3 and 4 are relatively prime, and the factor 5 is common. Therefore, LCM (15; 20) = 5 3 4 = 60.

Now we bring the fractions to common denominators:

Note how helpful factoring the original denominators was:

1. Having found the same factors, we immediately arrived at the least common multiple, which, generally speaking, is a nontrivial problem;
2. From the resulting expansion, you can find out which factors are "missing" for each of the fractions. For example, 234 3 = 702, therefore, for the first fraction, the additional factor is 3.

To estimate how colossal gains the least common multiple method gives, try calculating the same examples using the criss-cross method. Without a calculator, of course. I think after that comments will be superfluous.

Do not think that such complex fractions will not be in the real examples. They meet all the time, and the above tasks are not the limit!

The only problem is how to find this very NOC. Sometimes everything is found in a few seconds, literally "by eye", but on the whole this is a complex computational problem that requires separate consideration. We will not touch on this here.

See also:

Common denominator of fractions

Initially, I wanted to include common denominator methods in the Adding and Subtracting Fractions paragraph. But there was so much information, and its importance is so great (after all, common denominators are not only for numeric fractions) that it is better to study this issue separately.

So, let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The basic property of a fraction comes to the rescue, which, recall, sounds like this:

The fraction will not change if its numerator and denominator are multiplied by the same nonzero number.

Thus, if you choose the right factors, the denominators of the fractions will become equal - this process is called. And the required numbers, "leveling" the denominators, are called.

Why do you even need to bring fractions to a common denominator? Here are just a few reasons:

1. Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
2. Comparison of fractions. Sometimes converting to a common denominator makes this task much easier;
3. Solving problems for shares and percentages. Percentages are, in fact, common expressions that contain fractions.

There are many ways to find numbers that, when multiplied by, make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, efficiency.

Cross-multiplication

The easiest and most reliable way that is guaranteed to align the denominators. We will go ahead: we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators.

Take a look:

Task. Find the values ​​of the expressions:

Consider the denominators of neighboring fractions as additional factors. We get:

Yes, it's that simple. If you are just starting to learn fractions, it is better to work with this particular method - this way you will insure yourself against many mistakes and are guaranteed to get the result.

The only drawback of this method is that you have to count a lot, because the denominators are multiplied "ahead of time", and as a result, very large numbers can be obtained. This is the price to pay for reliability.

Common divisors method

This technique helps to greatly reduce the calculations, but, unfortunately, it is rarely used. The method is as follows:

1. Before you go ahead (that is, the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided by the other.
2. The number obtained as a result of such division will be an additional factor for the fraction with a lower denominator.
3. In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is the savings. At the same time, the probability of error is sharply reduced.

Task. Find the values ​​of the expressions:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is evenly divisible by the other, we apply the common factors method. We have:

Note that the second fraction was never multiplied by anything at all. In fact, we have cut the amount of computation in half!

By the way, I took the fractions in this example for a reason. If you're curious, try counting them crosswise. After the reduction, the answers will be the same, but there will be much more work.

This is the strength of the method of common divisors, but, I repeat, it can be applied only when one of the denominators is divisible by the other without a remainder. Which is rare enough.

Least Common Multiple Method

When we bring fractions to a common denominator, we are essentially trying to find a number that is divisible by each of the denominators. Then we bring the denominators of both fractions to this number.

There are a lot of such numbers, and the smallest of them will not necessarily be equal to the direct product of the denominators of the original fractions, as it is assumed in the "criss-cross" method.

For example, for the denominators 8 and 12, the number 24 is quite suitable, since 24: 8 = 3; 24: 12 = 2. This number is much less than the product 8 · 12 = 96.

The smallest number that is divisible by each of the denominators is called their (LCM).

Notation: the least common multiple of a and b is denoted by LCM (a; b). For example, LCM (16; 24) = 48; LCM (8; 12) = 24.

If you can find such a number, the total amount of computation will be minimal. Take a look at examples:

Task. Find the values ​​of the expressions:

Note that 234 = 117 · 2; 351 = 117 · 3. The factors 2 and 3 are coprime (they have no common divisors except 1), and the factor 117 is common. Therefore, the LCM (234; 351) = 117 · 2 · 3 = 702.

Similarly, 15 = 5 · 3; 20 = 5 · 4. The factors 3 and 4 are relatively prime, and the factor 5 is common. Therefore, LCM (15; 20) = 5 3 4 = 60.

Now we bring the fractions to common denominators:

Note how helpful factoring the original denominators was:

1. Having found the same factors, we immediately arrived at the least common multiple, which, generally speaking, is a nontrivial problem;
2. From the resulting expansion, you can find out which factors are "missing" for each of the fractions. For example, 234 3 = 702, therefore, for the first fraction, the additional factor is 3.

To estimate how colossal gains the least common multiple method gives, try calculating the same examples using the criss-cross method. Without a calculator, of course. I think after that comments will be superfluous.

Do not think that such complex fractions will not be in the real examples. They meet all the time, and the above tasks are not the limit!

The only problem is how to find this very NOC. Sometimes everything is found in a few seconds, literally "by eye", but on the whole this is a complex computational problem that requires separate consideration. We will not touch on this here.

See also:

Common denominator of fractions

Initially, I wanted to include common denominator methods in the Adding and Subtracting Fractions paragraph. But there was so much information, and its importance is so great (after all, common denominators are not only for numeric fractions) that it is better to study this issue separately.

So, let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The basic property of a fraction comes to the rescue, which, recall, sounds like this:

The fraction will not change if its numerator and denominator are multiplied by the same nonzero number.

Thus, if you choose the right factors, the denominators of the fractions will become equal - this process is called. And the required numbers, "leveling" the denominators, are called.

Why do you even need to bring fractions to a common denominator? Here are just a few reasons:

1. Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
2. Comparison of fractions. Sometimes converting to a common denominator makes this task much easier;
3. Solving problems for shares and percentages. Percentages are, in fact, common expressions that contain fractions.

There are many ways to find numbers that, when multiplied by, make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, efficiency.

Cross-multiplication

The easiest and most reliable way that is guaranteed to align the denominators. We will go ahead: we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators. Take a look:

Task. Find the values ​​of the expressions:

Consider the denominators of neighboring fractions as additional factors. We get:

Yes, it's that simple. If you are just starting to learn fractions, it is better to work with this particular method - this way you will insure yourself against many mistakes and are guaranteed to get the result.

The only drawback of this method is that you have to count a lot, because the denominators are multiplied "ahead of time", and as a result, very large numbers can be obtained.

Common denominator of fractions

This is the price to pay for reliability.

Common divisors method

This technique helps to greatly reduce the calculations, but, unfortunately, it is rarely used. The method is as follows:

1. Before you go ahead (that is, the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided by the other.
2. The number obtained as a result of such division will be an additional factor for the fraction with a lower denominator.
3. In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is the savings. At the same time, the probability of error is sharply reduced.

Task. Find the values ​​of the expressions:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is evenly divisible by the other, we apply the common factors method. We have:

Note that the second fraction was never multiplied by anything at all. In fact, we have cut the amount of computation in half!

By the way, I took the fractions in this example for a reason. If you're curious, try counting them crosswise. After the reduction, the answers will be the same, but there will be much more work.

This is the strength of the method of common divisors, but, I repeat, it can be applied only when one of the denominators is divisible by the other without a remainder. Which is rare enough.

Least Common Multiple Method

When we bring fractions to a common denominator, we are essentially trying to find a number that is divisible by each of the denominators. Then we bring the denominators of both fractions to this number.

There are a lot of such numbers, and the smallest of them will not necessarily be equal to the direct product of the denominators of the original fractions, as it is assumed in the "criss-cross" method.

For example, for the denominators 8 and 12, the number 24 is quite suitable, since 24: 8 = 3; 24: 12 = 2. This number is much less than the product 8 · 12 = 96.

The smallest number that is divisible by each of the denominators is called their (LCM).

Notation: the least common multiple of a and b is denoted by LCM (a; b). For example, LCM (16; 24) = 48; LCM (8; 12) = 24.

If you can find such a number, the total amount of computation will be minimal. Take a look at examples:

Task. Find the values ​​of the expressions:

Note that 234 = 117 · 2; 351 = 117 · 3. The factors 2 and 3 are coprime (they have no common divisors except 1), and the factor 117 is common. Therefore, the LCM (234; 351) = 117 · 2 · 3 = 702.

Similarly, 15 = 5 · 3; 20 = 5 · 4. The factors 3 and 4 are relatively prime, and the factor 5 is common. Therefore, LCM (15; 20) = 5 3 4 = 60.

Now we bring the fractions to common denominators:

Note how helpful factoring the original denominators was:

1. Having found the same factors, we immediately arrived at the least common multiple, which, generally speaking, is a nontrivial problem;
2. From the resulting expansion, you can find out which factors are "missing" for each of the fractions. For example, 234 3 = 702, therefore, for the first fraction, the additional factor is 3.

To estimate how colossal gains the least common multiple method gives, try calculating the same examples using the criss-cross method. Without a calculator, of course. I think after that comments will be superfluous.

Do not think that such complex fractions will not be in the real examples. They meet all the time, and the above tasks are not the limit!

The only problem is how to find this very NOC. Sometimes everything is found in a few seconds, literally "by eye", but on the whole this is a complex computational problem that requires separate consideration. We will not touch on this here.

When adding and subtracting algebraic fractions with different denominators, the fractions first lead to common denominator... This means that they find such a single denominator, which is divided by the original denominator of each algebraic fraction that is part of this expression.

As you know, if the numerator and denominator of a fraction are multiplied (or divided) by the same nonzero number, then the value of the fraction will not change. This is the basic property of a fraction. Therefore, when fractions lead to a common denominator, they essentially multiply the original denominator of each fraction by the missing factor to a common denominator. In this case, it is necessary to multiply by this factor and the numerator of the fraction (for each fraction it has its own).

For example, given the following amount algebraic fractions:

It is required to simplify the expression, that is, add two algebraic fractions. To do this, first of all, it is necessary to bring the terms-fractions to a common denominator. The first step is to find a monomial that is divisible by both 3x and 2y. In this case, it is desirable that it be the smallest, that is, find the least common multiple (LCM) for 3x and 2y.

For numerical coefficients and variables, the LCM is sought separately. LCM (3, 2) = 6 and LCM (x, y) = xy. Then the found values ​​are multiplied: 6xy.

Now we need to determine what factor to multiply 3x to get 6xy:
6xy ÷ 3x = 2y

This means that when reducing the first algebraic fraction to a common denominator, its numerator must be multiplied by 2y (the denominator has already been multiplied when reducing to a common denominator). The multiplier for the numerator of the second fraction is searched in a similar way. It will be equal to 3x.

Thus, we get:

Further, you can already act as with fractions with the same denominators: the numerators are added, and one common is written in the denominator:

After the transformations, a simplified expression is obtained, which is one algebraic fraction, which is the sum of the two original ones:

Algebraic fractions in the original expression can contain denominators, which are polynomials rather than monomials (as in the example above). In this case, before finding a common denominator, you should factor the denominators (if possible). Further, the common denominator is collected from different factors. If the factor is in several initial denominators, then it is taken once. If the factor has different degrees in the original denominators, then it is taken with a larger one. For example:

Here the polynomial a 2 - b 2 can be represented as the product (a - b) (a + b). The factor 2a - 2b is expanded as 2 (a - b). So the common denominator will be 2 (a - b) (a + b).

The denominator of the arithmetic fraction a / b is the number b, which shows the sizes of the unit fractions that make up the fraction. The denominator of the algebraic fraction A / B is called the algebraic expression B. To perform arithmetic operations with fractions, they must be reduced to the lowest common denominator.

You will need

• To work with algebraic fractions when finding the lowest common denominator, you need to know the methods of factoring polynomials.

Instructions

Consider the reduction to the lowest common denominator of two arithmetic fractions n / m and s / t, where n, m, s, t are integers. It is clear that these two fractions can be reduced to any denominator divisible by m and t. But they try to bring them to the lowest common denominator. It is equal to the least common multiple of the denominators m and t of these fractions. The least multiple (LCM) of numbers is the smallest that is divisible by all of the given numbers at the same time. Those. in our case it is necessary to find the least common multiple of the numbers m and t. It is designated as LCM (m, t). Then the fractions are multiplied by the corresponding ones: (n / m) * (LCM (m, t) / m), (s / t) * (LCM (m, t) / t).

Let's find the lowest common denominator of three fractions: 4/5, 7/8, 11/14. First, let's expand the denominators 5, 8, 14: 5 = 1 * 5, 8 = 2 * 2 * 2 = 2 ^ 3, 14 = 2 * 7. Next, we calculate the LCM (5, 8, 14), multiplying all the numbers included in at least one of the expansions. LCM (5, 8, 14) = 5 * 2 ^ 3 * 7 = 280. Note that if a factor occurs in the expansion of several numbers (factor 2 in the expansion of the denominators 8 and 14), then we take the factor in to a greater extent(2 ^ 3 in our case).

So, the total is received. It is 280 = 5 * 56 = 8 * 35 = 14 * 20. Here we get the numbers by which we need to multiply the fractions with the corresponding denominators in order to bring them to the lowest common denominator. We get 4/5 = 56 * (4/5) = 224/280, 7/8 = 35 * (7/8) = 245/280, 11/14 = 20 * (11/14) = 220/280.

Algebraic fractions are reduced to the lowest common denominator by analogy with arithmetic ones. For clarity, consider the problem by an example. Let two fractions (2 * x) / (9 * y ^ 2 + 6 * y + 1) and (x ^ 2 + 1) / (3 * y ^ 2 + 4 * y + 1) be given. Factor both denominators. Note that the denominator of the first fraction is a perfect square: 9 * y ^ 2 + 6 * y + 1 = (3 * y + 1) ^ 2. For

How to find the LCM (least common multiple)

A common multiple of two integers is an integer that is evenly divisible by both given numbers.

The least common multiple of two integers is the smallest integer divisible by both given numbers.

Method 1... You can find the LCM, in turn, for each of the given numbers, writing out in ascending order all the numbers that are obtained by multiplying them by 1, 2, 3, 4, and so on.

Example for numbers 6 and 9.
We multiply the number 6, sequentially, by 1, 2, 3, 4, 5.
We get: 6, 12, 18 , 24, 30
We multiply the number 9, sequentially, by 1, 2, 3, 4, 5.
We get: 9, 18 , 27, 36, 45
As you can see, the LCM for numbers 6 and 9 will be 18.

This method is convenient when both numbers are small and easy to multiply by a sequence of integers. However, there are times when you need to find the LCM for two-digit or three-digit numbers and also when the original numbers are three or even more.

Method 2... You can find the LCM by expanding the original numbers into prime factors.
After the expansion, it is necessary to delete the same numbers from the resulting series of prime factors. The remaining numbers of the first number will be a factor for the second, and the remaining numbers of the second will be a factor for the first.

Example for the number 75 and 60.
The least common multiple of 75 and 60 can be found without writing out the multiples of these numbers in a row. To do this, we decompose 75 and 60 into prime factors:
75 = 3 * 5 * 5, a
60 = 2 * 2 * 3 * 5 .
As you can see, factors 3 and 5 are found in both lines. Mentally we "cross out" them.
Let us write out the remaining factors included in the decomposition of each of these numbers. When expanding the number 75, we have the number 5 left, and when expanding the number 60, we have 2 * 2
So, to determine the LCM for the numbers 75 and 60, we need to multiply the remaining numbers from the decomposition of 75 (this is 5) by 60, and the numbers remaining from the decomposition of the number 60 (this is 2 * 2) multiply by 75. That is, for ease of understanding , we say that we are multiplying "crosswise".
75 * 2 * 2 = 300
60 * 5 = 300
This is how we found the LCM for the numbers 60 and 75. This is the number 300.

Example... Determine the LCM for numbers 12, 16, 24
In this case, our actions will be somewhat more complicated. But, first, as always, let us decompose all the numbers into prime factors
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3
To correctly determine the LCM, we choose the smallest of all numbers (this is the number 12) and sequentially go through its factors, crossing them out if at least one of the other series of numbers contains the same, not yet crossed out factor.

Step 1 . We see that 2 * 2 occurs in all rows of numbers. Cross them out.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3

Step 2. In the prime factors of the number 12, only the number 3 remains. But it is present in the prime factors of the number 24. Cross out the number 3 from both rows, while for the number 16 no action is assumed.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3

As you can see, when expanding the number 12, we "crossed out" all the numbers. This means that the finding of the NOC is completed. It remains only to calculate its value.
For the number 12, we take the remaining factors of the number 16 (the closest in ascending order)
12 * 2 * 2 = 48
This is the NOC

As you can see, in this case, finding the LCM was somewhat more difficult, but when you need to find it for three or more numbers, this way allows you to do it faster. However, both methods of finding the LCM are correct.

Let's continue talking about the least common multiple, which we started in the section "LCM - Least Common Multiple, Definition, Examples". In this topic, we will look at ways to find the LCM for three numbers or more, we will analyze the question of how to find the LCM of a negative number.

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Calculating the least common multiple (LCM) in terms of gcd

We have already established the relationship between the least common multiple and the greatest common divisor. Now we will learn how to determine the LCM in terms of the GCD. Let's first figure out how to do this for positive numbers.

Definition 1

Find the least common multiple of the greatest common divisor can be by the formula LCM (a, b) = a b: gcd (a, b).

Example 1

Find the LCM of numbers 126 and 70.

Solution

Let's take a = 126, b = 70. Substitute the values ​​in the formula for calculating the least common multiple through the greatest common divisor LCM (a, b) = a b: GCD (a, b).

Finds gcd of numbers 70 and 126. For this we need Euclid's algorithm: 126 = 70 1 + 56, 70 = 56 1 + 14, 56 = 14 4, therefore, GCD (126 , 70) = 14 .

We calculate the LCM: LCM (126, 70) = 126 70: GCD (126, 70) = 126 70: 14 = 630.

Answer: LCM (126, 70) = 630.

Example 2

Find the knock of numbers 68 and 34.

Solution

GCD in this case is not difficult, since 68 is divisible by 34. We calculate the least common multiple using the formula: LCM (68, 34) = 68 34: GCD (68, 34) = 68 34: 34 = 68.

Answer: LCM (68, 34) = 68.

In this example, we used the rule of finding the least common multiple for positive integers a and b: if the first number is divisible by the second, the LCM of these numbers will be equal to the first number.

Finding the LCM using prime factorization

Now let's look at a way to find the LCM, which is based on factoring numbers into prime factors.

Definition 2

To find the least common multiple, we need to perform a number of simple steps:

• compose the product of all prime factors of the numbers for which we need to find the LCM;
• we exclude all prime factors from the obtained products;
• the product obtained after eliminating common prime factors will be equal to the LCM of these numbers.

This method of finding the least common multiple is based on the equality LCM (a, b) = a b: GCD (a, b). If you look at the formula, it becomes clear: the product of the numbers a and b is equal to the product of all factors that are involved in the decomposition of these two numbers. In this case, the GCD of two numbers is equal to the product of all prime factors that are simultaneously present in the factorizations of these two numbers.

Example 3

We have two numbers, 75 and 210. We can factor them as follows: 75 = 3 5 5 and 210 = 2 3 5 7... If you compose the product of all factors of the two original numbers, you get: 2 3 3 5 5 5 7.

If we exclude the factors 3 and 5 common for both numbers, we get a product of the following form: 2 3 5 5 7 = 1050... This product will be our LCM for numbers 75 and 210.

Example 4

Find the LCM of numbers 441 and 700 by expanding both numbers into prime factors.

Solution

Let's find all the prime factors of the numbers given in the condition:

441 147 49 7 1 3 3 7 7

700 350 175 35 7 1 2 2 5 5 7

We get two chains of numbers: 441 = 3 · 3 · 7 · 7 and 700 = 2 · 2 · 5 · 5 · 7.

The product of all the factors that participated in the decomposition of these numbers will have the form: 2 2 3 3 5 5 7 7 7... Find the common factors. This number is 7. Let's exclude it from the general work: 2 2 3 3 5 5 7 7... It turns out that the NOC (441, 700) = 2 2 3 3 5 5 7 7 = 44 100.

Answer: LCM (441, 700) = 44 100.

Let us give one more formulation of the method for finding the LCM by decomposing numbers into prime factors.

Definition 3

Previously, we excluded from the total number of factors common to both numbers. Now we will do it differently:

• Let's decompose both numbers into prime factors:
• add the missing factors of the second number to the product of prime factors of the first number;
• we get the product, which will be the desired LCM of two numbers.

Example 5

Let's go back to the numbers 75 and 210, for which we already looked for the LCM in one of the previous examples. Let's decompose them into prime factors: 75 = 3 5 5 and 210 = 2 3 5 7... To the product of factors 3, 5 and 5 the number 75 add the missing factors 2 and 7 number 210. We get: 2 · 3 · 5 · 5 · 7. This is the LCM of numbers 75 and 210.

Example 6

Calculate the LCM of numbers 84 and 648.

Solution

Let us decompose the numbers from the condition into prime factors: 84 = 2 2 3 7 and 648 = 2 2 2 3 3 3 3... Add to the product the factors 2, 2, 3 and 7 number 84 missing factors 2, 3, 3 and
3 number 648. We get the work 2 2 2 3 3 3 3 7 = 4536. This is the least common multiple of 84 and 648.

Answer: LCM (84, 648) = 4,536.

Finding the LCM of three or more numbers

Regardless of how many numbers we are dealing with, the algorithm of our actions will always be the same: we will sequentially find the LCM of two numbers. There is a theorem for this case.

Theorem 1

Suppose we have integers a 1, a 2,…, a k... NOC m k of these numbers is found by sequentially calculating m 2 = LCM (a 1, a 2), m 3 = LCM (m 2, a 3),…, m k = LCM (m k - 1, a k).

Now let's look at how you can apply the theorem to solve specific problems.

Example 7

Calculate the least common multiple of four numbers 140, 9, 54, and 250 .

Solution

Let us introduce the notation: a 1 = 140, a 2 = 9, a 3 = 54, a 4 = 250.

Let's start by calculating m 2 = LCM (a 1, a 2) = LCM (140, 9). We apply Euclid's algorithm to calculate the GCD of numbers 140 and 9: 140 = 9 15 + 5, 9 = 5 1 + 4, 5 = 4 1 + 1, 4 = 1 4. We get: GCD (140, 9) = 1, LCM (140, 9) = 140 9: GCD (140, 9) = 140 9: 1 = 1 260. Therefore, m 2 = 1,260.

Now we calculate by the same algorithm m 3 = LCM (m 2, a 3) = LCM (1 260, 54). In the course of calculations, we get m 3 = 3 780.

It remains for us to calculate m 4 = LCM (m 3, a 4) = LCM (3 780, 250). We follow the same algorithm. We get m 4 = 94,500.

The LCM of the four numbers from the example condition is 94500.

Answer: LCM (140, 9, 54, 250) = 94,500.

As you can see, the calculations are simple, but rather laborious. To save time, you can go the other way.

Definition 4

We offer you the following algorithm of actions:

• decompose all numbers into prime factors;
• to the product of the factors of the first number, add the missing factors from the product of the second number;
• add the missing factors of the third number to the product obtained at the previous stage, etc .;
• the resulting product will be the least common multiple of all numbers from the condition.

Example 8

It is necessary to find the LCM of five numbers 84, 6, 48, 7, 143.

Solution

Let us decompose all five numbers into prime factors: 84 = 2 2 3 7, 6 = 2 3, 48 = 2 2 2 2 2 3, 7, 143 = 11 13. Prime numbers, which is the number 7, cannot be decomposed into prime factors. Such numbers coincide with their prime factorization.

Now take the product of prime factors 2, 2, 3 and 7 of 84 and add the missing factors of the second number to them. We split the number 6 into 2 and 3. These factors are already in the product of the first number. Therefore, we omit them.

We continue to add the missing factors. We pass to the number 48, from the product of prime factors of which we take 2 and 2. Then add a prime factor of 7 of the fourth number and factors of 11 and 13 for the fifth. We get: 2 2 2 2 3 7 11 13 = 48 048. This is the least common multiple of the original five numbers.

Answer: LCM (84, 6, 48, 7, 143) = 48,048.

Finding the Least Common Multiple of Negative Numbers

To find the least common multiple negative numbers, these numbers must first be replaced by numbers with the opposite sign, and then the calculations must be carried out according to the above algorithms.

Example 9

LCM (54, - 34) = LCM (54, 34) and LCM (- 622, - 46, - 54, - 888) = LCM (622, 46, 54, 888).

Such actions are permissible due to the fact that if we accept that a and - a- opposite numbers,
then the set of multiples a matches the set of multiples - a.

Example 10

It is necessary to calculate the LCM of negative numbers − 145 and − 45 .

Solution

Let's replace the numbers − 145 and − 45 on opposite numbers 145 and 45 ... Now, according to the algorithm, we calculate the LCM (145, 45) = 145 45: GCD (145, 45) = 145 45: 5 = 1 305, having previously determined the GCD according to the Euclidean algorithm.

We get that the LCM of the numbers is 145 and − 45 equals 1 305 .

Answer: LCM (- 145, - 45) = 1,305.

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