If J is divisible by 12 and 10, is j divisible by 24

Hi guys: This question is from the Number Properties book (5th ed) from Manhattan Guide. My answer is different from that stated in the solutions, so I am wondering if the book is incorrect.

So here is the question. I greatly appreciate the help as I'm about to smash my head into the wall. Thanks!

For questions #1-6, answer each question YES, NO, or CANNOT BE DETERMINED. If your answer
is CANNOT BE DETERMINED, use two numerical examples to show how the problem could
go either way. All variables in problems #1-6 are assumed to be integers.

4. If j is divisible by 12 and 10, is j divisible by 24?

Keep in mind that when you say a is divisible by b, it means 'b' is a factor of 'a'.

i.e. a = n*b
If b = 12, a can be 12/24/36/48/60.... etc

If j is divisible by 12, it means j has 12 as a factor (so j has at least two 2s and a 3 since 12 = 2*2*3).
If j is divisible by 10, it means j has 10 as a factor too (so j has at least one 2 - we already know that from above - and a 5).

So we know that j has at least two 2s, one 3 and one 5. To make 24, we need three 2s and one 3 (because 24 = 2*2*2*3). Do we know whether j has three 2s? No we don't.
j could be 2*2*2*3*5*7
it could be 2*2*3*5*7*11
or it could be 2*2*3*3*5*13
or many other things.
All we know is that it must have two 2s and one 3 and one 5. But this is not sufficient information.

Hey guys,

Just starting to get into quant studies and I came across a stumper. The question states, "if j is divisible by 12 and 10, is j divisible by 24"

I created prime boxes for 12 and 10 and deduced it had to be since 3x2x2x2 = 24.

However my Manhattan GMAT book states it cannot be determined since there are only two 2's that are definitely in the prime factorization of j because the 2 in the prime factorization of 10 may be redundant as one of the 2's in the prime factorization of 12.

I have to admit, I don't follow that at all. Any assistance would be greatly appreciated.

Thanks!

I should note, I can't believe how much math I have forgotten. I need the basic refresher imaginable...

Merging similar topics. Please go through the posts above and ask if anything remains unclear.

Generally if positive integer j is divisible by 12 and 10, then it must be divisible by the least common multiple of 12 and 10, so by 60. This implies that j can be 60, 120, 180, ... Now, if j=60, then it won't be divisible by 24 but if it's for example, 120 then it will be.

The following DS question might also help: is-j-divisible-by-138043.html

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