Question

Challenge  You are given the​ numbers, StartSet 38 plus n comma StartFraction n Over 3 EndFraction comma StartRoot n plus 16 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

We need 38 + n, n/3, and sqrt(n + 16) all to be natural numbers. So n/3 must be an integer ⇒ n is a multiple of 3, and n + 16 must be a perfect square. Let n + 16 = k^2, so n = k^2 − 16.

Try the smallest k making n positive: k = 5 gives n = 25 − 16 = 9. This is a multiple of 3, and
38 + 9 = 47 (natural), n/3 = 3, sqrt(n + 16) = 5.

Thus the smallest such n is 9.