What two numbers have a GCF of 25 and a LCM of 900?

Answer 1

Factorise 25 and 900 into their prime factorisations:

25 = 5²

900 = 2² x 3² x 5²

With a GCF of 25 = 5², that leaves 2² × 3² which must be distributed between the two numbers so that there are no common primes. There are two ways to do this:

1. All of 2² × 3² is in one of the numbers, making the two numbers 5² = 25 and 2² x 3² x 5² = 900
2. Put the prime 2 in one number and the prime 3 in the other, making the two numbers: 5² × 2² = 100 and 5² × 3² = 225

The two possible numbers are:

• 25 and 900; or
• 100 and 225.

Answer 2

If all three numbers are different: 1, 5, 25.

Otherwise, additionally one of the sets: {1, 1, 25}, {1, 25, 25}, {5, 5, 25}, {5, 25, 25}, {25, 25, 25}.

Answer 3

5 and 25