The integer 2006 is the product of three prime numbers p, q, and r. If p + q = r^2 (p + r), what is the value of q?

2006 = 2•17•59 (prime factorization)
so p q rare permutions of 2,17,59
now r^2 divides p+q so 17 and 59 are too large for r so r= 2
so p+q = 4 p + 8
or 3p = q - 8
so p < q-8 => so p=17 and 3p = 51 and q = 59 satisfying the condition
hence q = 59