2002 AMC 12A Problem 12

Attempt Problem 12 of the 2002 AMC 12A 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 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

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

If the roots are primes

p

and

q,

then

p + q = 63

An odd sum forces one of the primes to be

2

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.

Problem 12 in other years