ReMO-e

Number Theory Level 3

Find the number of positive integers $n<2015$ such that $n(2015-n)$ is a multiple of 2015.

1 solution

Siddharth Singh
Nov 2, 2015

The question says:

$n(2015-n)=2015k$ (for some integer k)

$2015n-2015k=n^{2}$

$(n-k)=\frac{n^{2}}{2015}$

As L.H.S is an integer , $2015|n^{2}$ consequently $2015|n$ (Since $2015=5*13*31$ )

This implies $2015 \le n$ ,but the question says $n<2015$

Hence, there are no such positive integers.

Good solution !!

Akshat Sharda - 5 years, 7 months ago