Problem:
Several sets of prime numbers, such as , use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
Answer Choices:
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Solution:
First, observe that 4,6 , and 8 cannot be the units digit of any two-digit prime, so they must contribute at least to the sum. The remaining digits must contribute at least to the sum. Thus, the sum must be at least 207, and we can achieve this minimum only if we can construct a set of three one-digit primes and three two-digit primes. Using the facts that nine is not prime and neither two nor five can be the units digit of any two-digit prime, we can construct the sets , , or , each of which yields a sum of .
The problems on this page are the property of the MAA's American Mathematics Competitions