MP guide 5 chapter 1 problem 4

[MichaelJ353]

Question: If j is divisible by 12 and 10, is j divisible by 24?
The answer on page 25 says cannot be determined. Maybe I am doing something really stupid, but can't 12 and 10 be broken down into (2 x 2 x 3) and (2 x 5)? And then you can multiply (2 x 2 x 3 x 2) = 24? Meaning yes, j is divisible by 24.

[Sage Pearce-Higgins]

Analyzing the prime factors is a great way to solve this, but there's a catch! Sure, we know that j is divisible by 10 and that, in other words, means j has a 2 and a 5 in its prime factors. And the same for 12: we know that j has two 2s and a 3 in its prime factors. But do we know that j has three 2s in its prime factors? Definitely not, since we could "reuse" one of the twos.

Looking from another perspective, can you find the two cases that show insufficiency? I.e. find one number for j that is divisible by 24 and one number that isn't. Post them here when you've done it.