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=\frac{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.