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Problem 983

AMC 12 late, AIME early

Number theory Difficulty 4.8 Prove it AMC 10 A · United States

Let $n$ be the least prime number that can be written as the sum of 5 distinct prime numbers. What is the sum of the digits of $n$ ?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

The prime $2$ cannot be among the $5$ distinct primes chosen because, if it were, then the sum would be even. The first $5$ odd primes are $3$ , $5$ , $7$ , $11$ , and $13$ , and their sum is $39$ , which is not prime. The next smallest sum of $5$ distinct odd primes is $3 + 5 + 7 + 11 + 17 = 43$ , which is prime. The requested digit sum is $4 + 3 = 7$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.