If $5 \mid n (n^2 + 1) (n + 1) (n - 1)$, why can $n$ be of the form $5k$, $5k + 1$, $5k + 2$, $5k + 3$, $5k + 4$? Why can't it be $5k + 5$, $5k + 6$, etc?
Asked
Active
Viewed 40 times
-1
-
Consider division $n$ by $5$. It gives some remainder $0\le m\le 4$ and quotient $k$, thus $n=5k+m$. You may as well consider $m\ge 5$, but considering $0\le m<5$ is sufficent (exhaustive). – Alexey Burdin Aug 18 '20 at 00:25
-
If $n=5k+5$ then $n=5K$ with $K=k+1$, etc. – J. W. Tanner Aug 18 '20 at 00:31
1 Answers
3
If $n = 5k+5 = 5(k+1)$ then let $m=k+1$ and $n=5m$.
Similarly, if $n = 5k+6 = 5(k+1)+1$, again let $m=k+1$ and $n=5m+1$.
The idea is, when you divide $a \div b$, then the remainder must be one of $0,1,\ldots, b-1$. Otherwise, you did not divide correctly.
gt6989b
- 54,422