How easy it is to find the common denominator of two numbers. Ways to find the least common multiple, nok is, and all explanations

Consider three ways to find the least common multiple.

Finding by Factoring

The first way is to find the least common multiple by factoring the given numbers into prime factors.

Suppose we need to find the LCM of numbers: 99, 30 and 28. To do this, we decompose each of these numbers into prime factors:

For the desired number to be divisible by 99, 30 and 28, it is necessary and sufficient that it includes all the prime factors of these divisors. To do this, we need to take all the prime factors of these numbers to the highest occurring power and multiply them together:

2 2 3 2 5 7 11 = 13 860

So LCM (99, 30, 28) = 13,860. No other number less than 13,860 is evenly divisible by 99, 30, or 28.

To find the least common multiple of given numbers, you need to decompose them into prime factors, then take each prime factor with the largest exponent with which it occurs, and multiply these factors together.

Since coprime numbers have no common prime factors, then their least common multiple is equal to the product of these numbers. For example, three numbers: 20, 49 and 33 are coprime. So

LCM (20, 49, 33) = 20 49 33 = 32,340.

The same should be done when looking for the least common multiple of various prime numbers. For example, LCM (3, 7, 11) = 3 7 11 = 231.

Finding by selection

The second way is to find the least common multiple by fitting.

Example 1. When the largest of the given numbers is evenly divisible by other given numbers, then the LCM of these numbers is equal to the larger of them. For example, given four numbers: 60, 30, 10 and 6. Each of them is divisible by 60, therefore:

NOC(60, 30, 10, 6) = 60

In other cases, to find the least common multiple, the following procedure is used:

Determine the largest number from the given numbers.
Next, find numbers that are multiples the largest number, multiplying it by integers in ascending order and checking whether the remaining given numbers are divisible by the resulting product.

Example 2. Given three numbers 24, 3 and 18. Determine the largest of them - this is the number 24. Next, find the numbers that are multiples of 24, checking whether each of them is divisible by 18 and by 3:

24 1 = 24 is divisible by 3 but not divisible by 18.

24 2 = 48 - divisible by 3 but not divisible by 18.

24 3 \u003d 72 - divisible by 3 and 18.

So LCM(24, 3, 18) = 72.

Finding by Sequential Finding LCM

The third way is to find the least common multiple by successively finding the LCM.

The LCM of two given numbers is equal to the product of these numbers divided by their greatest common divisor.

Example 1. Find the LCM of two given numbers: 12 and 8. Determine their greatest common divisor: GCD (12, 8) = 4. Multiply these numbers:

We divide the product into their GCD:

So LCM(12, 8) = 24.

To find the LCM of three or more numbers, the following procedure is used:

First, the LCM of any two of the given numbers is found.
Then, the LCM of the found least common multiple and the third given number.
Then, the LCM of the resulting least common multiple and the fourth number, and so on.
Thus the LCM search continues as long as there are numbers.

Example 2. Find the LCM three data numbers: 12, 8 and 9. The LCM of the numbers 12 and 8 we have already found in the previous example (this is the number 24). It remains to find the least common multiple of 24 and the third given number - 9. Determine their greatest common divisor: gcd (24, 9) = 3. Multiply LCM with the number 9:

We divide the product into their GCD:

So LCM(12, 8, 9) = 72.

When adding and subtracting algebraic fractions with different denominators first the fractions 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 number other than zero, then the value of the fraction will not change. This is the main property of a fraction. Therefore, when fractions lead to a common denominator, in fact, the original denominator of each fraction is multiplied 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 (it is different for each fraction).

For example, given the following sum of algebraic fractions:

It is required to simplify the expression, i.e., add two algebraic fractions. To do this, first of all, it is necessary to reduce 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, i.e., find the least common multiple (LCM) for 3x and 2y.

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

Now we need to determine by what factor we need 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 reduced to a common denominator). The factor for the numerator of the second fraction is similarly searched for. It will be equal to 3x.

Thus, we get:

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

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

Algebraic fractions in the original expression may contain denominators that are polynomials rather than monomials (as in the above example). In this case, before finding a common denominator, 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 multiplier 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 a product (a - b)(a + b). The factor 2a – 2b is expanded as 2(a – b). Thus, the common denominator will be equal to 2(a - b)(a + b).

To solve examples with fractions, you need to be able to find the smallest common denominator. Below is a detailed instruction.

How to find the lowest common denominator - concept

Least common denominator (LCD) in simple words is the smallest number that is divisible by the denominators of all fractions this example. In other words, it is called the Least Common Multiple (LCM). NOZ is used only if the denominators of the fractions are different.

How to find the lowest common denominator - examples

Let's consider examples of finding NOZ.

Calculate: 3/5 + 2/15.

Solution (Sequence of actions):

We look at the denominators of fractions, make sure that they are different and the expressions are reduced as much as possible.
We find lesser number, which is divisible by both 5 and 15. This number will be 15. Thus, 3/5 + 2/15 = ?/15.
We figured out the denominator. What will be in the numerator? An additional multiplier will help us figure this out. An additional factor is the number obtained by dividing the NOZ by the denominator of a particular fraction. For 3/5, the additional factor is 3, since 15/5 = 3. For the second fraction, the additional factor is 1, since 15/15 = 1.
Having found out the additional factor, we multiply it by the numerators of the fractions and add the resulting values. 3/5 + 2/15 = (3*3+2*1)/15 = (9+2)/15 = 11/15.

Answer: 3/5 + 2/15 = 11/15.

If in the example not 2, but 3 or more fractions are added or subtracted, then the NOZ must be searched for as many fractions as given.

Calculate: 1/2 - 5/12 + 3/6

Solution (sequence of actions):

Finding the lowest common denominator. The minimum number divisible by 2, 12 and 6 is 12.
We get: 1/2 - 5/12 + 3/6 = ?/12.
We are looking for additional multipliers. For 1/2 - 6; for 5/12 - 1; for 3/6 - 2.
We multiply by the numerators and assign the corresponding signs: 1/2 - 5/12 + 3/6 = (1 * 6 - 5 * 1 + 2 * 3) / 12 = 7/12.

Answer: 1/2 - 5/12 + 3/6 = 7/12.

Multiplication "criss-cross"

Common divisor method

Task. Find expression values:

To appreciate how much of a win the least common multiple method gives, try calculating the same examples using the criss-cross method.

Common denominator of fractions

Of course, without a calculator. I think after that comments will be redundant.

See also:

I originally wanted to include the common denominator methods in the "Adding and Subtracting Fractions" paragraph. But there turned out to be so much information, and its importance is so great (after all, not only 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 main property of a fraction comes to the rescue, which, let me remind you, sounds like this:

A fraction does not change if its numerator and denominator are multiplied by the same non-zero number.

Thus, if you choose the factors correctly, the denominators of the fractions will be equal - this process is called. And the desired numbers, "leveling" the denominators, are called.

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

Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
Fraction comparison. Sometimes reduction to a common denominator greatly simplifies this task;
Solving problems on shares and percentages. Percentages are, in fact, ordinary expressions that contain fractions.

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

Multiplication "criss-cross"

The simplest and most reliable way, which is guaranteed to equalize the denominators. We will act "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 original denominators. Take a look:

Task. Find expression values:

As additional factors, consider the denominators of neighboring fractions. We get:

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

The only downside this method- you have to count a lot, because the denominators are multiplied "throughout", and as a result you can get very big numbers. That's the price of reliability.

Common divisor method

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

Look at the denominators before you go "thru" (i.e., "criss-cross"). Perhaps one of them (the one that is larger) is divisible by the other.
The number resulting from such a division will be an additional factor for a fraction with a smaller denominator.
At the same time, 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 expression values:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divisible by the other without a remainder, we apply the method of common factors. We have:

Note that the second fraction was not multiplied by anything at all. In fact, we have cut the amount of calculations in half!

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

This is the strength of the method. common divisors, but, I repeat, it can only be used if one of the denominators is divided by the other without a remainder. Which happens quite rarely.

Least common multiple method

When we reduce 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 equal the direct product of the denominators of the original fractions, as is assumed in the "cross-wise" method.

For example, for denominators 8 and 12, the number 24 is quite suitable, since 24: 8 = 3; 24:12 = 2. This number is much less 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 numbers a and b is denoted by LCM(a; b). For example, LCM(16; 24) = 48; LCM(8; 12) = 24.

If you manage to find such a number, the total amount of calculations will be minimal. Look at the examples:

How to find the lowest common denominator

Find expression values:

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 LCM(234; 351) = 117 2 3 = 702.

Similarly, 15 = 5 3; 20 = 5 4. Factors 3 and 4 are coprime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.

Now let's bring the fractions to common denominators:

Note how useful the factorization of the original denominators turned out to be:

Discovering same multipliers, we immediately reached the least common multiple, which, generally speaking, is a non-trivial problem;
From the resulting expansion, you can find out which factors are “missing” for each of the fractions. For example, 234 3 \u003d 702, therefore, for the first fraction, the additional factor is 3.

Don't think that these complex fractions in the real examples will not. They meet all the time, and the above tasks are not the limit!

The only problem is how to find this NOC. Sometimes everything is found in a few seconds, literally “by eye”, but in general this is a complex computational problem that requires separate consideration. Here we will not touch on this.

See also:

Bringing fractions to a common denominator

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

A fraction does not change if its numerator and denominator are multiplied by the same non-zero number.

Thus, if you choose the factors correctly, the denominators of the fractions will be equal - this process is called. And the desired numbers, "leveling" the denominators, are called.

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

Common denominator, concept and definition.

Here are just a few reasons:

Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
Fraction comparison. Sometimes reduction to a common denominator greatly simplifies this task;
Solving problems on shares and percentages. Percentages are, in fact, ordinary expressions that contain fractions.

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

Multiplication "criss-cross"

Task. Find expression values:

As additional factors, consider the denominators of neighboring fractions. We get:

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

Common divisor method

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

Look at the denominators before you go "thru" (i.e., "criss-cross"). Perhaps one of them (the one that is larger) is divisible by the other.
The number resulting from such a division will be an additional factor for a fraction with a smaller denominator.
At the same time, 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 expression values:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divisible by the other without a remainder, we apply the method of common factors. We have:

Note that the second fraction was not multiplied by anything at all. In fact, we have cut the amount of calculations in half!

This is the strength of the method of common divisors, but, again, it can only be applied when one of the denominators is divided by the other without a remainder. Which happens quite rarely.

Least common multiple method

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

For example, for 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 of 8 12 = 96.

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

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

If you manage to find such a number, the total amount of calculations will be minimal. Look at the examples:

Task. Find expression values:

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 LCM(234; 351) = 117 2 3 = 702.

Similarly, 15 = 5 3; 20 = 5 4. Factors 3 and 4 are coprime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.

Now let's bring the fractions to common denominators:

Note how useful the factorization of the original denominators turned out to be:

Having found the same factors, we immediately reached the least common multiple, which, generally speaking, is a non-trivial problem;
From the resulting expansion, you can find out which factors are “missing” for each of the fractions. For example, 234 3 \u003d 702, therefore, for the first fraction, the additional factor is 3.

To appreciate how much of a win the least common multiple method gives, try calculating the same examples using the criss-cross method. Of course, without a calculator. I think after that comments will be redundant.

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

See also:

Bringing fractions to a common denominator

A fraction does not change if its numerator and denominator are multiplied by the same non-zero number.

Thus, if you choose the factors correctly, the denominators of the fractions will be equal - this process is called. And the desired numbers, "leveling" the denominators, are called.

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

Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
Fraction comparison. Sometimes reduction to a common denominator greatly simplifies this task;
Solving problems on shares and percentages. Percentages are, in fact, ordinary expressions that contain fractions.

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

Multiplication "criss-cross"

Take a look:

Task. Find expression values:

As additional factors, consider the denominators of neighboring fractions. We get:

Common divisor method

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

Look at the denominators before you go "thru" (i.e., "criss-cross"). Perhaps one of them (the one that is larger) is divisible by the other.
The number resulting from such a division will be an additional factor for a fraction with a smaller denominator.
At the same time, 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 expression values:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divisible by the other without a remainder, we apply the method of common factors. We have:

Note that the second fraction was not multiplied by anything at all. In fact, we have cut the amount of calculations in half!

This is the strength of the method of common divisors, but, again, it can only be applied when one of the denominators is divided by the other without a remainder. Which happens quite rarely.

Least common multiple method

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

For example, for 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 of 8 12 = 96.

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

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

If you manage to find such a number, the total amount of calculations will be minimal. Look at the examples:

Task. Find expression values:

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 LCM(234; 351) = 117 2 3 = 702.

Similarly, 15 = 5 3; 20 = 5 4. Factors 3 and 4 are coprime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.

Now let's bring the fractions to common denominators:

Note how useful the factorization of the original denominators turned out to be:

Having found the same factors, we immediately reached the least common multiple, which, generally speaking, is a non-trivial problem;
From the resulting expansion, you can find out which factors are “missing” for each of the fractions. For example, 234 3 \u003d 702, therefore, for the first fraction, the additional factor is 3.

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

See also:

Bringing fractions to a common denominator

A fraction does not change if its numerator and denominator are multiplied by the same non-zero number.

Thus, if you choose the factors correctly, the denominators of the fractions will be equal - this process is called. And the desired numbers, "leveling" the denominators, are called.

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

Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
Fraction comparison. Sometimes reduction to a common denominator greatly simplifies this task;
Solving problems on shares and percentages. Percentages are, in fact, ordinary expressions that contain fractions.

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

Multiplication "criss-cross"

Task. Find expression values:

As additional factors, consider the denominators of neighboring fractions. We get:

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

Bringing fractions to a common denominator

That's the price of reliability.

Common divisor method

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

Look at the denominators before you go "thru" (i.e., "criss-cross"). Perhaps one of them (the one that is larger) is divisible by the other.
The number resulting from such a division will be an additional factor for a fraction with a smaller denominator.
At the same time, 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 expression values:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divisible by the other without a remainder, we apply the method of common factors. We have:

Note that the second fraction was not multiplied by anything at all. In fact, we have cut the amount of calculations in half!

This is the strength of the method of common divisors, but, again, it can only be applied when one of the denominators is divided by the other without a remainder. Which happens quite rarely.

Least common multiple method

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

For example, for 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 of 8 12 = 96.

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

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

If you manage to find such a number, the total amount of calculations will be minimal. Look at the examples:

Task. Find expression values:

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 LCM(234; 351) = 117 2 3 = 702.

Similarly, 15 = 5 3; 20 = 5 4. Factors 3 and 4 are coprime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.

Now let's bring the fractions to common denominators:

Note how useful the factorization of the original denominators turned out to be:

Having found the same factors, we immediately reached the least common multiple, which, generally speaking, is a non-trivial problem;
From the resulting expansion, you can find out which factors are “missing” for each of the fractions. For example, 234 3 \u003d 702, therefore, for the first fraction, the additional factor is 3.

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

To bring fractions to the least common denominator, you must: 1) find the least common multiple of the denominators of these fractions, it will be the least common denominator. 2) find an additional factor for each of the fractions, for which we divide the new denominator by the denominator of each fraction. 3) multiply the numerator and denominator of each fraction by its additional factor.

Examples. Reduce the following fractions to the lowest common denominator.

We find the least common multiple of the denominators: LCM(5; 4) = 20, since 20 is the smallest number that is divisible by both 5 and 4. We find for the 1st fraction an additional factor 4 (20 : 5=4). For the 2nd fraction, the additional multiplier is 5 (20 : 4=5). We multiply the numerator and denominator of the 1st fraction by 4, and the numerator and denominator of the 2nd fraction by 5. We reduced these fractions to the lowest common denominator ( 20 ).

The lowest common denominator of these fractions is 8, since 8 is divisible by 4 and itself. There will be no additional multiplier to the 1st fraction (or you can say that it equal to one), to the 2nd fraction the additional factor is 2 (8 : 4=2). We multiply the numerator and denominator of the 2nd fraction by 2. We reduced these fractions to the lowest common denominator ( 8 ).

These fractions are not irreducible.

We reduce the 1st fraction by 4, and we reduce the 2nd fraction by 2. ( see examples for abbreviations ordinary fractions: Sitemap → 5.4.2. Examples of reduction of ordinary fractions). Find LCM(16 ; 20)=2 4 · 5=16· 5=80. The additional multiplier for the 1st fraction is 5 (80 : 16=5). The additional multiplier for the 2nd fraction is 4 (80 : 20=4). We multiply the numerator and denominator of the 1st fraction by 5, and the numerator and denominator of the 2nd fraction by 4. We reduced these fractions to the lowest common denominator ( 80 ).

Find the least common denominator of the NOC(5 ; 6 and 15) = LCM(5 ; 6 and 15)=30. The additional multiplier to the 1st fraction is 6 (30 : 5=6), the additional multiplier to the 2nd fraction is 5 (30 : 6=5), the additional multiplier to the 3rd fraction is 2 (30 : 15=2). We multiply the numerator and denominator of the 1st fraction by 6, the numerator and denominator of the 2nd fraction by 5, the numerator and denominator of the 3rd fraction by 2. We reduced these fractions to the lowest common denominator ( 30 ).

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