2002 AMC 10A Problem 14

Attempt Problem 14 of the 2002 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 10A solutions, or check the answer key.

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14.

Both roots of the quadratic equation

x^2-63x+k=0

are prime numbers. The number of possible values of

k

0

1

2

4

more than four

The roots sum to

63

and multiply to

k

An odd sum of two primes forces one of them to be

2

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.

k=2\cdot 61=122

is the only possible value.

Thus, the correct answer is B.

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