Problem 20 · 2023 Math Kangaroo
Hard
Number Theory
primessum-constraint
The arithmetic mean of five different prime numbers is an integer. What is the smallest possible value of this arithmetic mean?
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Answer: B — 6
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Hint 1 of 2
The mean being an integer means the five primes sum to a multiple of 5.
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Hint 2 of 2
Start from the smallest primes and adjust to hit a multiple of 5 while keeping the mean small.
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Approach: minimise the average under a divisibility constraint
- Five different primes sum to at least 2+3+5+7+11 = 28, so the mean is at least 5.6.
- The mean must be a whole number, so the smallest possible is 6 (sum 30).
- Indeed 2+3+5+7+13 = 30 works, giving mean 6.
- So the smallest mean is 6.
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