Question

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?

$21$

$60$

$119$

$180$

$231$

Answer: C

Concepts:prime parity factoring

Difficulty rating: 1310

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 it is $3 \pmod 4.$ This leaves $119$ and $231.$ The latter would require $(x-1)(y-1)=232,$ but no two distinct numbers in $\{4,6,10,12,16\}$ have product $232.$

Indeed, $\begin{aligned} 11 \cdot 13 - (11 + 13) &= 143 - 24 \\ &= 119. \end{aligned}$

Thus, the correct answer is C.

2000 AMC 12 Problem 6

Answer: C

Difficulty rating: 1310

Solution:

Problem 6 in Other Years