2002 AMC 10A Problem 14

Problem 14 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

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Solution

If the roots are primes

p

and

q,

then

p+q=63

and

pq=k.

Because

63

is odd, one prime must be

2,

making the other

61,

which is prime. So

k=2\cdot 61=122

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

Concepts: Vieta’s Formulas · prime · parity

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