Question

Using prime factorization, how can you determine whether the least common multiple of two numbers is the product of the two numbers, or is less than the product of the two numbers? Options: Option 1: By comparing the sum of prime factors to the product of the numbers. Option 2: By comparing the highest common factor to the product of the numbers. Option 3: By comparing the prime factors to the sum of the two numbers. Option 4: By comparing the prime factors to the difference of the two numbers.

Answer 1

Explanation:

the LCM is the product of the longest chains per prime factor.

e.g. when looking at 12 and 30 :

12 = 2×2×3

30 = 2×3×5

the longest chain of 2s is 2×2.

the longest chain of 3s is 3.

the longest chain of 5s is 5.

the LCM of 12 and 30 = 2×2×3×5 = 60

the highest (or greatest) common factor GCF is the product of the prime factors they have in common :

2 and 3 for 12 and 30, so the GCF = 2×3 = 6.

the LCM is the product of both numbers, if they have no prime factors in common (there is no "shortcut" between them). there is no GCF (besides the standard "1").

e.g.

11 and 49

11 = 11

49 = 7×7

the longest chain of 7 is 7×7.

the longest chain of 11 is 11.

the LCM of 11 abs 49 = 7×7×11 = 11×49 = 539

the GCF = 1 (as there are no prime factors they have in common).

so, clearly, the correct answer is

option 2 : comparing the GCF to the product of the numbers.

comparing products with sums (also a difference is a sum, just with a negative number) does not make any sense here.