Question

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Part 1
Challenge  You are given the​ numbers, StartSet 30 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 n/3 to be an integer and √(n+16) to be an integer. Let √(n+16)=m (an integer ≥4), so n = m^2 − 16. We also need m^2 − 16 to be divisible by 3.

m = 4 → n = 0 (0/3 = 0, √(0+16)=4). If 0 is not considered a natural number, continue:
m = 5 → n = 25 − 16 = 9, and 9/3 = 3, √(9+16)=5.

Thus the smallest positive n making all entries natural is 9. (If 0 is allowed as natural, the smallest n would be 0.)