Asked by estrella
Challenge You are given the numbers, StartSet 35 plus n comma StartFraction n Over 5 EndFraction comma StartRoot n plus 144 EndRoot EndSet
. Find the smallest value of n so that all of the numbers in the set are natural numbers.
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Part 1
The smallest value of n that makes all of the numbers in set natural numbers is
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All Answers 1
Answered by
GPT-5 mini
AI
We need n/5 to be an integer, so n is a multiple of 5, and n + 144 must be a perfect square.
Let n = 5m. Then n + 144 = k^2, i.e. k^2 − 144 = 5m. The smallest k ≥ 12 giving a positive multiple of 5 is k = 13, since 13^2 − 144 = 169 − 144 = 25. Thus n = 25.
Check: 35 + 25 = 60, 25/5 = 5, √(25 + 144) = 13 — all natural numbers.
Answer: 25.
Let n = 5m. Then n + 144 = k^2, i.e. k^2 − 144 = 5m. The smallest k ≥ 12 giving a positive multiple of 5 is k = 13, since 13^2 − 144 = 169 − 144 = 25. Thus n = 25.
Check: 35 + 25 = 60, 25/5 = 5, √(25 + 144) = 13 — all natural numbers.
Answer: 25.
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