The answer is A
From stmt 1, p = (t+m)/2; and hence p must be even integer; and hence mpt must be even
From stmt 2; t and m both could be either even or odd and no info on P and hence not sufficient

GmatKiss wrote:If m, p, and t are positive integers and m<p<t, is the product mpt an even integer?

(1) t - p = p - m
(2) t - m = 16

[spoiler]IMO:A[/spoiler]

For mpt to be odd, all 3 values (m, p and t) must be odd.

Question rephrased: Are m, p and t all odd?

Statement 1: t-p = p-m.
m+t = 2p
p = (m+t)/2.
The implication is that p is HALFWAY between m and t.
Thus, all 3 values could be even (2,4,6), or all 3 values could be odd (1,3,5).
INSUFFICIENT.

Statement 2: t = m+16.
If m=1, then t=17.
If p=9 , then all 3 values are odd.
If p=10, then all 3 values are not odd.
INSUFFICIENT.

Statements 1 and 2 combined:
To satisfy both statements, t = m+16, and p is halfway between m and t.
If m=1 and t=17, then p=9, in which case all 3 values are odd.
If m=2 and t=18, then p=10, in which case all 3 values are not odd.
INSUFFICIENT.

The correct answer is E.