2000 AMC 12 Problem 6

Below is the professionally curated solution for Problem 6 of the 2000 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2000 AMC 12 solutions, or check the answer key.

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Concepts:prime parity factoring

Difficulty rating: 1310

Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

$21$

$60$

$119$

$180$

$231$

Solution:

The primes between $4$ and $18$ are $5, 7, 11, 13,$ and $17.$

For two such primes, $xy - (x + y) = (x - 1)(y - 1) - 1$ is a product of two even numbers minus $1,$ hence odd. Among the choices, only $119$ is odd.

Indeed, $11 \cdot 13 - (11 + 13) = 143 - 24 = 119.$

Thus, the correct answer is C.

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