How to Find the Least Common Multiple of Two Numbers

Multiple is the product of a number with an integer. The least common multiple of a group of numbers is the smallest number that is divisible by all of them. To find the least common multiple, you need to determine the factor of each number. There are several different methods of finding the least common multiple, and they also work for three or more numbers.

Table of Contents

List multiples

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Review your numbers. This method is suitable for cases where two numbers need to find a common multiple less than 10. For larger numbers, you should use another method.

Take for example the problem of finding the least common multiple of 5 and 8. Since both numbers are small, it is well suited to use this method.

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List the first few multiples of the first number. Multiple is the product of a number with an integer. ^{[1] X Research Sources} In other words, they are numbers that appear in your times table.

For example, the first multiples of 5 are 5, 10, 15, 20, 25, 30, 35, and 40, respectively.

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List the first few multiples of the second number. You should write near the list of multiples of the first number for easy comparison.

For example, the first multiples of 8 include 8, 16, 24, 32, 40, 48, 56, and 64.

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Find the least common multiple of the numbers above. You will probably have to write more to the list of multiples until you find a number that is both a multiple of one and a multiple of another. That is your least common multiple. ^{[2] X Research Source}

For example, 40 is the smallest number that satisfies the condition that it is both a multiple of 5 and a multiple of 8, so the least common multiple of 5 and 8 is 40.

Factoring out primes

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Consider your numbers. This method is suitable for numbers greater than 10. For smaller numbers, you can use another method to find the least common multiple more quickly.

For example, to find the least common multiple of 20 and 84, you should use this method.

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Analyze the first number. Here we will decompose this number into prime factors, that is, find prime numbers whose product is equal to the given number. To do that, we can use tree diagrams. After the analysis is complete, we will rewrite it as an equation.

For example: $2$ $\times$ $ten$ $=$ $20$ ${displaystyle mathbf {2} times 10=20}$ ${mathbf {2}}times 10=20$ and $2$ $\times$ $5$ $=$ $ten$ ${displaystyle mathbf {2} times mathbf {5} =10}$ ${mathbf {2}}times {mathbf {5}}=10$ , so the prime factors of 20 are 2, 2 and 5. Rewritten as an equation, we have: $20$ $=$ $2$ $\times$ $2$ $\times$ $5$ ${displaystyle 20=2times 2times 5}$ $20=2times 2times 5$ .

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Analyze the second number. As with the first number, we find the prime factors whose product is equal to the second.

For example: $2$ $\times$ $42$ $=$ $84$ ${displaystyle mathbf {2} times 42=84}$ ${mathbf {2}}times 42=84$ , $7$ $\times$ $6$ $=$ $42$ ${displaystyle mathbf {7} times 6=42}$ ${mathbf {7}}times 6=42$ , and $3$ $\times$ $2$ $=$ $6$ ${displaystyle mathbf {3} times mathbf {2} =6}$ ${mathbf {3}}times {mathbf {2}}=6$ , so the prime factors of 84 are 2, 7, 3, and 2. Rewrite we get $84$ $=$ $2$ $\times$ $7$ $\times$ $3$ $\times$ $2$ ${displaystyle 84=2times 7times 3times 2}$ $84=2times 7times 3times 2$ .

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Write down the common factors. Multiply common factors. Cross out each common factor in the analytic equation to prime each time it is drawn.

For example, both numbers have a factor of 2, so we write $2$ $\times$ ${displaystyle 2times }$ $2times$ and cross out a 2 in both factoring equations.
Both numbers also share another factor of 2, so we’ll add $2$ $\times$ $2$ ${displaystyle 2times 2}$ $2times 2$ and cross out the second factor of 2 in each of the original analytical equations.

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Add all remaining factors to the multiplication. Those are the factors that are not crossed out after you have compared the two groups of factors. They are unshared factors. ^{[3] X Research Sources}

For example in the equation $20$ $=$ $2$ $\times$ $2$ $\times$ $5$ ${displaystyle 20=2times 2times 5}$ $20=2times 2times 5$ , we crossed out both 2s because they are also in the other. And since there’s 5 left, we’ll add in the multiplication: $2$ $\times$ $2$ $\times$ $5$ ${displaystyle 2times 2times 5}$ $2times 2times 5$ .
In the equation $84$ $=$ $2$ $\times$ $7$ $\times$ $3$ $\times$ $2$ ${displaystyle 84=2times 7times 3times 2}$ $84=2times 7times 3times 2$ , we also crossed out both 2s. With 7 and 3 left, we’ll add in the multiplication: $2$ $\times$ $2$ $\times$ $5$ $\times$ $7$ $\times$ $3$ ${displaystyle 2times 2times 5times 7times 3}$ $2times 2times 5times 7times 3$ .

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Calculate the least common multiple. To do that, we just need to multiply the numbers in the multiplication we just made.

For example: $2$ $\times$ $2$ $\times$ $5$ $\times$ $7$ $\times$ $3$ $=$ $420$ ${displaystyle 2times 2times 5times 7times 3=420}$ $2times 2times 5times 7times 3=420$ . So the least common multiple of 20 and 84 is 420.

Use the grid or ladder method

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Draw a checkered grid. The checkered grid consists of two sets of parallel lines perpendicular to each other. They form three columns and look like the pound sign (#) on a phone or keyboard. Write the first number in the top, center box. Write the second number in the top right box. ^{[4] X Research Sources}

For example, for the problem of finding the least common multiple of 18 and 30, we write 18 in the upper position, in the center of the grid and 30 in the upper right.

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Find some common factor of both numbers. Write this number in the top left box. Although not required, it is better if the factor is a prime factor.

In the example problem, since 18 and 30 are even, 2 is their common factor. Therefore, we will write 2 in the upper left cell of the grid.

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Divide each number by the factor you just found and write the quotient in the box below. The quotient is the result of division.

Because $18$ $\div$ $2$ $=$ $9$ ${displaystyle 18div 2=9}$ $18div 2=9$ so 9 will be written under 18.
$30$ $\div$ $2$ $=$ $15$ ${displaystyle 30div 2=15}$ $30div 2=15$ , so 15 is written under 30.

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Find the common factor of the two quotients. If there are no more common factors, you can skip and go to the next step. If there is a common factor, we write it in the middle left cell of the grid.

For example, 9 and 15 are both divisible by 3, so we will write 3 in the middle left cell of the grid.

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Divide the quotient by this common factor. Write a new spear below the head injury.

$9$ $\div$ $3$ $=$ $3$ ${displaystyle 9div 3=3}$ $9div 3=3$ so 3 will be written below 9.
$15$ $\div$ $3$ $=$ $5$ ${displaystyle 15div 3=5}$ $15div 3=5$ so 5 will be written below 15.

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Expand the grid if needed. Continue in this way until the two quotients have no common factor.

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Circle the numbers on the first and last columns of the grid, forming an “L”. Set to multiply all these factors. ^{[5] X Research Sources}

For example, since 2 and 3 are in the first column and 3 and 5 are in the last row, we have $2$ $\times$ $3$ $\times$ $3$ $\times$ $5$ ${displaystyle 2times 3times 3times 5}$ $2times 3times 3times 5$ .

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Complete the multiplication. When we multiply these numbers together, we get the least common multiple of the two given numbers. ^{[6] X Research Source}

Eg $2$ $\times$ $3$ $\times$ $3$ $\times$ $5$ $=$ $90$ ${displaystyle 2times 3times 3times 5=90}$ $2times 3times 3times 5=90$ . Therefore, 90 is the least common multiple of 18 and 30.

Using Euclidean algorithm

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Understand the terminology used in division. The divisor is the number given to be divided. The divisor is the number by which the divisor is divided. The quotient is the answer to division. The remainder is the remainder after the division is completed. ^{[7] X Research Sources}

For example in the equation $15$ $\div$ $6$ $=$ $2$ ${displaystyle 15div 6=2}$ $15div 6=2$ residual $3$ ${displaystyle 3}$ $3$ :
15 is the divisor
6 is divisor
2 is lovable
3 is the balance.

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Set up the quotient-residual formula. That is: divisor = divisor x quotient + remainder. ^{[8] X Research Source} You will use it to set up the Euclidean algorithm to find the greatest common divisor of two given numbers.

Eg $15$ $=$ $6$ $\times$ $2$ $+$ $3$ ${displaystyle 15=6times 2+3}$ $15=6times 2+3$ .
The greatest common divisor is the divisor, or greatest factor, of both numbers. ^{[9] X Research Source}
In this method, we will first find the greatest common divisor and then use it to find the least common multiple.

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Take the larger number as the divisor and the smaller number as the divisor. Set up a quotient-remainder equation for these two numbers.

For example, for the problem of finding the least common multiple of 210 and 45, we will calculate $210$ $=$ $45$ $\times$ $4$ $+$ $30$ ${displaystyle 210=45times 4+30}$ $210=45times 4+30$ .

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Take the original divisor as the new divisor, and the initial remainder as the new divisor. Set up a quotient-remainder equation for these two numbers.

For example: $45$ $=$ $30$ $\times$ $2$ $+$ $15$ ${displaystyle 45=30times 2+15}$ $45=30times 2+15$ .

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Repeat until the remainder is zero. For each new equation, we use the divisor of the previous equation as the divisor and the previous remainder as the divisor. ^{[10] X Research Source}

For example: $30$ $=$ $15$ $\times$ $2$ $+$ $0$ ${displaystyle 30=15times 2+0}$ $30=15times 2+0$ . Since the remainder is zero, we will stop here.

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Look at the final divisor. This is the greatest common divisor of the two original numbers. ^{[11] X Research Source}

In the example problem, since the last equation is $30$ $=$ $15$ $\times$ $2$ $+$ $0$ ${displaystyle 30=15times 2+0}$ $30=15times 2+0$ and the last divisor is 15 so 15 is the greatest common divisor of 210 and 45.

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Multiply two numbers. Divide the product by their greatest common divisor. The result obtained is the least common multiple of the two given numbers. ^{[12] X Research Source}

For example: $210$ $\times$ $45$ $=$ $9450$ ${displaystyle 210times 45=9450}$ $210times 45=9450$ . Divide by the greatest common divisor, we get: $9450$ $15$ $=$ $630$ ${displaystyle {frac {9450}{15}}=630}$ ${frac {9450}{15}}=630$ . So 630 is the least common multiple of 210 and 45.

Advice

To find the least common multiple of three or more numbers, we can modify the above methods slightly. For example, to find the least common multiple of 16, 20 and 32, you can find the least common multiple of 16 and 20 first (which is 80), and then find the least common multiple of 80 and 32 to get the result. the last is 160.
The least common multiple is frequently used. The most common is in the addition and subtraction of fractions: the fractions must have the same denominator, and therefore, if they have different denominators, you will have to reduce the denominator to perform the calculation. The best way is to find the least common denominator – which is the least common multiple of the denominators.

Steps

List multiples

Factoring out primes

Use the grid or ladder method

Using Euclidean algorithm

Advice

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