Divisiblilty & Prime Problems [GRE]

Question

if j is divisible by 12 and 10, is j divisible by 24?

Answer by either saying yes, no , or Can't be determined.

I approached this question as follow:

First i found the prime factors of both numbers

10: 5 * 2

12 : 3*2*2

Then i found out the prime factors of 24

24: 2*3*2*2

So i concluded from this since we need 3 2s and 1 3 the answer is yes.

However the book states that the answer is can't be determined because one of the 2s can be redundant. I am not sure about this reasoning can anyone explain me why? Also if you have some reliable source please present it.

Answer 1

A counter-example is the number 60. 60 is divisible by 10 and 12 but is not divisible by 24.

Suppose you have a number n which is divisible by both 10 and 12. This means that there must be at least 2 twos in the prime decomposition of n (based on the prime decomposition of 10 and 12). Here you can see that it is not necessarily true that there are 3 twos that divide n.

Answer 2

If $j$ is divisible by $12 = 2^2 \cdot 3$ and by $10 = 2 \cdot 5$, then we can certainly conclude it's divisible by $2^2 \cdot 3 \cdot 5$, and so by $2^2 \cdot 3 = 12$. We can't conclude that it's divisible by $2^3$, though, since neither $12$ nor $10$ are.

But it's not always true that it's divisible by $2^3$; the number $60$ is a counterexample.