Question

Challenge  You are given the​ numbers, StartSet 38 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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Answer 1

n must be a positive multiple of 5 and n+144 must be a perfect square. Let n+144 = k^2 (k>12), so n = k^2 − 144. We need k^2 ≡ 144 (mod 5) ⇒ k^2 ≡ 4 (mod 5), so k ≡ 2 or 3 (mod 5). The smallest k>12 with that property is k=13, giving n = 13^2 − 144 = 169 − 144 = 25.

Answer: 25.