<p>If p and n are integers such that p>n>0
and p^2 - n^2 = 12, which of the following
can be the value of p-n?</p>
<p>I. 1
II. 2
III. 4</p>
<p>A) I only
B) II only
C) I and II only
D) II and III only
E) I, II, and III</p>
<p>I simplified p^2 - n^2 = 12 into (p-n)(p+n) = 12
After that, trying to plug-in values seems to be very
tedious and inefficient.</p>
<p>If p-n = 4, then p+n = 3; I would try to partially solve
the systems of equation to see if I could get an integer value.</p>
<p>adding the equations together, you get 2p = 7, so p = 3.5, not an integer, so III is out of the question..</p>
<p>If p-n = 2, then p+n = 6 I would solve and get 2p = 8, p =4 an integer, so II works.</p>
<p>If p-n = 1, then p+n = 12. Solve and get 2p =11, p =5.5 not an integer, so I does not work.</p>
<p>Therefore, the answer is B. I felt my method was too tedious.
Any better suggested methods?</p>