Manhattan Challenge Problem of the week! - 7 march 07

The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer

(2) q is less than p

(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Answer - E

The product pq factors: {1, p, q, and pq}. If pq has no additional factors, then positive integers p and q must be prime. Pay heed to the fact that neither p nor q can be equal to 1; otherwise, pq would not have 4 distinct factors.

Statement (1) -- the sum of p and q is an odd integer. Therefore either p is even and q is odd, or p is odd and q is even. Since we know that both p and q are prime, either p or q must be equal to 2 (the only even prime number). But we do not know which of the two integers is equal to 2.

Statement (2) -- q is less than p -- gives us no information about the value of p

Taking both statements together, we can conclude that, because 2 is the smallest prime number, q must equal 2. But, we cannot determine the value of p.