If the sum of three integers is even, is the product of the three integers a multiple of 4 ?
(1) All three integers are equal.
(2) All three integers are even.

Let a + b + c is even and we want to a * b * c is a multiple of 4 or not!
Now if we take 1, a = b = c and again it has to be that a + b + c is even so that means a, b and c have to be even. So each of a, b and c is multiple of 2. So a * b * c should be multiple of 4.
According to 2, it is clearly saying that a, b and c are even so we can easily incorporate the above logic here.
So D.

If the sum of three integers is even, is the product of the three integers a multiple of 4 ?

(1) All three integers are equal.
so all need to even so that sum is even, hence divisible by 4 SUFF

(2) All three integers are even.

same as above

Is picking numbers a good strategy to use here??

Yop! U can also go with that, moneyman! It really depends whether u r able to break the Qs or not obvly in some limited time-period but not on the approach u r taking.