2000 AMC 12 Problem 6

Problem 6 of 25EasierAlgebra Number Theory

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?

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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.