2002 AMC 12A Problem 12 — Solution | LIVE by Po-Shen Loh

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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 $k$ is

$0$

$1$

$2$

$4$

more than four

Answer: B

Concepts:Vieta’s Formulas prime parity

Difficulty rating: 1350

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.

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