Question

PLEASE HELP ME IF YOU CAN!!!! Find all integer numbers n, such that 1+2+3+ ... +n is divisible by 5.

Answer 1

If

$S=1+2+\cdots+(n-1)+n$

then it's also true that

$S=n+(n-1)+\cdots+2+1$

so that

$2S=(1+n)+(2+(n-1))+\cdots+((n-1)+2)+(n+1)=n(n+1)\implies S=\dfrac{n(n+1)}2$

For any integer $n$, $n(n+1)$ is even. We have $5\mid S$ if either $5\mid n$ or $5\mid(n+1)$. This means $n$ must satisfy either $n\equiv0\pmod5$ or $n\equiv4\pmod5$, in which case we have a general solution of

$n=5k$

or

$n=5k-1$

where $k$ is any integer.