## 1995 Remainder

When an integer is divided by $1995$, the remainder is $75$. When the same integer is divided by $57$, what will be the remainder?
Source: mathcontest.olemiss.edu 2/6/2012

SOLUTION
Let $n$ be the integer. By definition of division remainder, there exists a quotient $q$ such that
$n=q\left (1995\right )+75$
$=q\left (35\times 57\right )+\left (57+18\right )$
Factor 57
$n=57\left (35q+1\right )+18$
Let $p=35q+1$
$n=p\left (57\right )+18$
The remainder equals $18$ when the same integer $n$ is divided by $57$.

Answer: $18$.