2000 AMC 10 Problem 11

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

The primes between

4

and

18

are

5,

7,

11,

13,

17,

all odd

Note that

xy - (x + y)

= (x - 1)(y - 1) - 1,

which is odd and increases with

x

and

y

The primes between

4

and

18

are

5,

7,

11,

13,

and

17.

The product of two of them is odd and the sum is even, so

xy - (x + y)

is odd.

Since

xy - (x + y)

= (x - 1)(y - 1) - 1

increases as either prime increases, the result ranges from

5 \cdot 7 - 12 = 23

up to

13 \cdot 17 - 30 = 191.

The only odd option in

[23, 191]

is

119 = 11 \cdot 13 - (11 + 13).

Thus, the correct answer is C.

Problem 11 in other years