2002 AMC 12A Problem 12

Problem 12 of 25IntermediateAlgebra Number Theory

Both roots of the quadratic equation

x^2 - 63x + k = 0

are prime numbers. The number of possible values of

k

Show solution

Solution

If the roots are primes

p

and

q,

then by Vieta’s formulas

p + q = 63

and

pq = k.

Since

63

is odd, one prime must be even, namely

2,

and the other is

61.

Both are prime, so

k = 2\cdot 61 = 122

is the only possible value. Thus, the correct answer is B.

Concepts: Vieta’s Formulas · prime · parity

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.