Question

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?

$193$

$207$

$225$

$252$

$477$

Answer: B

Concepts:prime digits optimization

Difficulty rating: 1800

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.

2002 AMC 12A Problem 17

Answer: B

Difficulty rating: 1800

Solution:

Problem 17 in Other Years