2002 AMC 12A Problem 17

Problem 17 of 25IntermediateAlgebra Number Theory

Several sets of prime numbers, such as

\{7, 83, 421, 659\},

use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?

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Solution

The even digits

4, 6, 8

cannot be the units digit of a multi-digit prime, so each must appear in a tens place or higher, contributing at least

40 + 60 + 80 = 180.

The other six digits contribute at least

1 + 2 + 3 + 5 + 7 + 9 = 27,

so the sum is at least

207.

This bound is achieved, for example by

\{2, 3, 5, 41, 67, 89\},

whose sum is

207.

Thus, the correct answer is B.