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. First I found the prime factors of both numbers (10: 5 * 2 and 12 : 3*2*2). Then I found out the prime factors of 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

"j is divisible by 12 and 10" means that there is some integer n such that j = 2*2*3*5*n.

"j is divisible by 24" means that there is some integer m such that j =2*2*2*3*m.

Consider n=1.so j = 60. j IS NOT divisible by 24.

Consider n=2 so j =120. j IS divisible by 24 with m = 5

Consider n=3 so j = 180 j IS NOT divisible by 24.

Consider n=4 so j = 240 j IS divisible by 24, with m=10

. . .

Consider n=10 and j = 600 j IS divisible by 24 with m = 25

because 2*2*3*5*n = 2*2*2*3*m

5*n = 2*m

has integer values of (n,m) for (2,5), (4,10), (6,15), . . .

The answer is "can't be determined." This is because n must be even (that is, n has a prime factor of 2).

This statement is always true:

If j is divisible by 24 and 20 then j is divisible by 24 (how obvious is that?).

Answer 2

What you know is that j has 2², 3 and 5 among its factors.

If j has no other factors, then the answer is NO, because 60 is NOT divisible by 24.

If j has another 2 among its factors, then the answer is YES.