Asked by josh
Challenge You are given the numbers, StartSet 40 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 + 144 = m^2, so n = m^2 − 144 with integer m ≥ 12.
m^2 − 144 ≡ 0 (mod 5) ⇒ m^2 ≡ 144 ≡ 4 (mod 5), so m ≡ 2 or 3 (mod 5). The smallest m ≥ 12 with that property is m = 12, giving n = 0, but 0 is usually not considered a natural number. The next is m = 13, giving n = 13^2 − 144 = 169 − 144 = 25.
Thus the smallest natural n is 25.
m^2 − 144 ≡ 0 (mod 5) ⇒ m^2 ≡ 144 ≡ 4 (mod 5), so m ≡ 2 or 3 (mod 5). The smallest m ≥ 12 with that property is m = 12, giving n = 0, but 0 is usually not considered a natural number. The next is m = 13, giving n = 13^2 − 144 = 169 − 144 = 25.
Thus the smallest natural n is 25.
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