Question

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

Answer 1

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.