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For how many ordered pairs of positive integers (x,y), with y are both (x)/(y) and (x+1)/(y+1) integers?

User Dan Short
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1 Answer

5 votes

Answer: Infinitely many

Explanation:

Let the value of
(x)/(y) be
k, where
k is an integer. Then,
x=ky.


\implies (x+1)/(y+1)=(ky+1)/(y+1)=(k(y+1))/(y+1)+(1-k)/(y+1)=k+(1-k)/(y+1)

This means we need
(1-k)/(y+1) to be an integer. If we let this fraction equal
n, then:


(1-k)/(y+1)=n \implies 1-k=ny+1 \implies y=(-k)/(n)

From this, we see that there are infinitely many pairs.

User Dcoles
by
8.3k points

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