Least Common Multiples by Prime Factorization

The unique factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number. The short term for Least common multiples is LCM. (Source Wikipedia)

Examples for finding Least common multiples by prime factorization:

Example 1:

Find the least common multiples for the numbers 234 and 534 by prime factorization.

Solution:

Prime factorization for the given numbers are

234   :  2 3 3 13

534   :  2 3        89

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LCM:  2 3 3 13 89

Least common multiples by prime factorization is 2 * 3 * 3 * 13 * 89 = 20826

Example 2:

Find the least common multiples for the numbers 584 and 564 by prime factorization.

Solution:

584    : 2 2 2 73

564    : 2 2 3 47

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LCM : 2 2 2 3 73 47

Least common multiples by prime factorization is 2 * 2 * 2 * 3 * 73 * 47 = 82344

Example 3:

Find the least common multiples for the numbers 124 and 164 by prime factorization.

Solution:

124    : 2 2 31

164    : 2 2 41

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LCM : 2 2 31 41

Least common multiples by prime factorization is 2 * 2 * 31 * 41 = 5084

Example 4:

Find the least common multiples for the numbers 48 and 58 by prime factorization.

Solution:

48      : 2 2 2 2 3

58      : 2            29

------------------------------

LCM : 2 2 2 2 3 29

Least common multiples by prime factorization is 2 * 2 * 2 * 2 * 3 * 29 = 1392

Example 5:

Find the least common multiples for the numbers 45 and 64 by prime factorization.

Solution:

45      : 5 3 3

64      :          2 2 2 2 2 2

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LCM : 5 3 3 2 2 2 2 2 2

Least common multiples by prime factorization is 5 * 3 * 3 * 2 * 2 * 2 * 2 * 2 * 2= 2880

Practice problems for finding Least common multiples by prime factorization:

Problem 1:

Find the least common multiples for the numbers 56 and 456 by prime factorization.

Least common multiples by prime factorization is 3192

Problem 2:

Find the least common multiples for the numbers 64 and 65 by prime factorization.

Least common multiples by prime factorization is 4160

Problem 3:

Find the least common multiples for the numbers 560 and 4560 by prime factorization.

Least common multiples by prime factorization is 31920

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Problem 4:

Find the least common multiples for the numbers 356 and 78 by prime factorization.

Least common multiples by prime factorization is 13884

Problem 5:

Find the least common multiples for the numbers 4562 and 5697 by prime factorization.

Least common multiples by prime factorization is 25989714