How do you check least common multiples by prime factorization?

Answer 1

You factor each of the two numbers (whose least common multiple your are looking for) down to all the prime numbers that multiply together to get those numbers.

Factoring Example: 36 = 6x6 = 2x3x2x3 = 2x2x3x3

Factoring Example: 7 = 7

Factoring Example: 15 = 3x5

Factoring Example 14 = 2x7

To get the least common multiple from the factors, you must look over all of the factors and take all of the UNIQUE ones with their highest multiples (like in 36 above, you have to take 2 twos and 2 threes) and multiply all of those together.

So... the least common multiple for 36 and 7 is 2x2x3x3x7 = 252

the least common multiple for 7 and 15 is 3x5x7 = 105

the least common multiple for 36 and 15 is 2x2x3x3x5 = 180

the least common multiple for 7 and 14 is 2x7 = 14

the least common multiple for 36 and 14 is 2x2x3x3x7= 252

Note: ALL of the factors of both numbers are in each answer, but no more than we need.

Keep in mind that LEAST common multiples are always equal to or GREATER than the larger of the two numbers. They are BIG numbers because they are multiples.

It is confusing when you learn GREATEST common factor, which is always LESS than or equal to the smaller of the two numbers. They are SMALL numbers because they are FACTORS.

Note: you can take least common multiples of more than two numbers:

Example: LCM of 14,15 & 36 is 2x2x3x3x5x7= 1260

If you write the prime factorizations of the numbers in power format, the lcm of the numbers is obtained by multiplying together the highest power of all the primes (across all the numbers).

eg lcm of 14, 15 & 36:

14 = 2 x 7

15 = 3 x 5

36 = 2^2 x 3^2

lcm = 2^2 x 3^2 x 5 x 7 = 1260

eg lcm 12, 14, 15:

12 = 2^2 x 3

14 = 2 x 7

15 = 3 x 5

lcm = 2^2 x 3 x 5 x 7 = 420

In this example, the highest power of 3 (which is 1) occurs in both 12 and 15, but it still only occurs once in the lcm.