Problem Solving

jsche229 wrote:If J is divisible by 12 and 10, is J divisible by 24?

For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)


So, if J is divisible by 12, then J = (2)(2)(3)(?)(?)(?)... [note: there could be other numbers in the prime factorization of J, thus the question marks]
Likewise, if J is divisible by 10, then J = (2)(5)(?)(?)...
Combine all of this and we can conclude that there are AT LEAST two 2's, one 3 and one 5 hiding in the prime factorization of J
So, it's possible that J = 60, since 60 = (2)(3)(3)(5). Notice that 60 is NOT divisible by 24.
It's also possible that J = 120, since 120 = (2)(2)(3)(3)(5). Notice that 120 IS divisible by 24

Cheers,
Brent

Hi jsche229,

Brent's solution focuses on the prime factorization of numbers. Here's another approach that you can use relatively quickly to assess your thinking:

J is divisible by both 10 and 12, so J is a MULTIPLE of 10 AND a MULTIPLE of 12

Multiples of 10 are easy: 10, 20, 30, 40, etc.
Multiples of 12: 12, 24, 36, 48, 60,......120, etc.

Notice that 60 is a multiple of 10 and 20? But 60 is NOT divisible by 24.
120 is also a multiple of 10 and 20. 120 IS divisible by 24.

The GMAT regularly tests the thoroughness of your thinking, so unless your math knowledge and "theory" are perfect, you'll have to do more work to get the correct answer.

GMAT assassins aren't born, they're made,
Rich