f2001290 wrote:k, m, and t are positive integers and k/6 + m/4 = 12t, do t and 12 have a common factor greater than 1 ?
(1) k is a multiple of 3.
(2) m is a multiple of 3.
We have 2 k + 3 m = 144 t, where k, m, and t are positive integers. The question otherwise is, "Are t and 12 not co prime?"
(1) If k is a multiple of 3, so is 144 or 12; hence, t may or may not have 3 as its factor. Insufficient.
(2) If m is a multiple of 3, 3 m is a multiple of 9, so is 144; hence, t may or may not have 3 as its factor. Insufficient.
When taken together, let's take k = 3 a and m = 3 b where a and b are positive integers, so that 6 a + 9 b = 144 t, or 2 a + 3 b = 48 t. Now, if a is a multiple of 3, t may or may not have 3 as its factor, otherwise t and 12 must have a common factor greater than 1.
[spoiler]E[/spoiler]
