Problem Solving

Hey all-

I am solving problem 5 in the MGMAT number properties book (12 ed). The problem is as follows:

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

The answer choices are yes, no and can't be determined.

The answer I keep getting is yes due to the prime factorization however the answer key says I am 1 "2" short. The reason is because the "2" when factoring 10 is a duplicate.

Can anyone help me understand why it is a duplicate? Thanks!

jsche229 wrote:Hey all-

I am solving problem 5 in the MGMAT number properties book (12 ed). The problem is as follows:

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

The answer choices are yes, no and can't be determined.

The answer I keep getting is yes due to the prime factorization however the answer key says I am 1 "2" short. The reason is because the "2" when factoring 10 is a duplicate.

Can anyone help me understand why it is a duplicate? Thanks!

Good question! This comes up a lot.

An easy way to "see" why this doesn't work is to consider the LCM of 12 and 10, which is 60. 60 divides by both 12 and 10, but it doesn't divide by 24. So dividing by both 12 and 10 is NOT enough to make a number divisible by 24.

A more abstract way to demonstrate this would be as follows.

If x is divisible by 12, then the prime factorization of x must contain 2 * 2 * 3 (at least two 2s and one 3).

If x is divisible by 10, then the prime factorization of x must contain 2 * 5 (at least one 2 and one 5).

If x is divisible by 12 and by 10, then we take everything we've got above: x is divisible by at least two 2s, at least one 3, and at least one 5. So x is divisible by 2 * 2 * 3 * 5, or 60.

Notice that the "extra" 2 from 10 doesn't matter: since x divides by 12, we know we already have at least one 2, so this simply overlaps with the 2 we find in 10.

That now makes sense. Would you apply the LCM method to solving problems or the abstract? Thanks!

jsche229 wrote:Hey all-

I am solving problem 5 in the MGMAT number properties book (12 ed). The problem is as follows:

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

The answer choices are yes, no and can't be determined.

The answer I keep getting is yes due to the prime factorization however the answer key says I am 1 "2" short. The reason is because the "2" when factoring 10 is a duplicate.

Can anyone help me understand why it is a duplicate? Thanks!

Find the LCM of 10 and 12 , it comes out to be 60

Now all multiples of 60 are divisible by both 10 and 12

So , the multiples of 60 are - 60 , 120 , 180 , 240 , 300 ........

No divide the first few terms to check if 24 is divided without leaving any remainder or not.

60/24 = Not divisible

120/24 = Divisible

180/24 = Not divisible


So we cant be certain , coz some numbers are divisible while some are not...