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Home » The number of ways is $\boxed243$. - Appcentric

The number of ways is $\boxed243$. - Appcentric

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 $ 14 \div 13 = 1 $ remainder 1 â yes.

📌 And 14 is a multiple of 7 â valid.

📌 Is there a smaller two-digit value? Next smaller would be $ m = -1 $: $ L = 91(-1) + 14 = -77 $, not positive.

📌 So 14 is the smallest.

📌 But wait: is 14 a valid total length made from 7-unit pipes? Yes: two pipes â $ 7 imes 2 = 14 $.

📌 Thus, the smallest two-digit total length satisfying the condition is $ oxed{14} $.

📌 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.

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