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April 11th, 2018, 01:29 AM   #1
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Is there a method to solve these types of questions?

With questions such as:

Find all values of n such that both expressions are integers

n and n+16/n+1
2n and 7/n

etc...

Is there a method in which this can be solved without trial and error?

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April 11th, 2018, 04:37 AM   #2
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No.

Let's take the second problem as an example.

Let m = 2n, so that m is required to be an integer.
As 7/n = 14/m is also required to be an integer, we know that m divides 14.
It follows (by trial and error) that m = ±1, ±2, ±7 or ±14.
Hence n = ±1/2, ±1, ±7/2 or ±7.
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