Competition problem — Number theory

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Number theory Difficulty 5.2 AIME, harder Find the answer

4. Ana chose the digits $1,2,3,4,5,6,7$ and 9. She decided to form groups of 4 prime numbers and use all the chosen digits for each group of prime numbers. What is the sum of the prime numbers in each group?

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48th Mathematical Competition

for high school students of Slovenia

Ljutomer, April 17, 2004

## Tasks for 2nd year students

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

I/4. Ana can form a group of prime numbers $\{23,41,59,67\}$ with the selected digits. To avoid searching for all possible groups of prime numbers, let's think differently. A two-digit prime number cannot end with any of the digits $2,4,5$ or 6, so these digits must appear in the tens place. No matter how Ana forms a group of 4 prime numbers, their sum is equal to $10 \cdot(2+4+5+$ 6) $+(1+3+7+9)=170+20=190$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.