Is the integer k divisible by 4?

(1) 8k is divisible by 16.
(2) 9k is divisible by 12.

best way to go, if you don't get the theory right away: start running through an exhaustive list of possibilities for each statement, and see if either
(a) you can prove the statement is insufficient (in which case you're done), or
(b) a pattern emerges that will allow you to conclude that the statement is sufficient.

statement (1): just consider an exhaustive list of multiples of 16
if 8k = 16, then k = 2 --> NO
if 8k = 32, then k = 4 --> YES
insufficient

statement (2): just consider an exhaustive list of multiples of 12
if 9k = 12, then k isn't an integer
if 9k = 24, then k isn't an integer
if 9k = 36, then k = 4 --> YES
if 9k = 48, then k isn't an integer
if 9k = 60, then k isn't an integer
if 9k = 72, then k = 8 --> YES
a pattern has emerged (continue a couple more rounds if you like)
sufficient

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For statement 2 :
* the prime factorization of 9k is 3 x 3 x (whatever goes into k)
* this factorization must contain 4, which is two 2's
* both of the 2's must be part of k, because the rest of the factorization is 3's
* therefore, k is divisible by 4