Check if 39 can be written as sum of distinct primes:

February 22, 2026

Check if 39 can be written as sum of distinct primes:

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Question: A primatologist observes that a troop of monkeys communicates using sequences of calls, each lasting a prime number of seconds. If the total duration of a sequence longer than 20 seconds but less than 40 seconds consists of the sum of distinct prime-numbered call durations, what is the largest possible total duration?
Solution: We seek the largest integer $ N $ such that $ 20 < N < 40 $, and $ N $ is the sum of distinct prime numbers.
We aim for the largest such $ N $, so start from 39 downward.
is odd â must include the only even prime, 2, or use odd primes only.
Try including 2: then we need sum of distinct odd primes â¡ 37 mod 2 â odd, OK.
So check: 39 = 2 + 37? Yes, 2 and 37 are distinct primes.

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