Question

Prove that for all integers n, n(n + 2)(n + 4) is divisible by 3.

Answer 1

Answer on Question #85800 – Math – Discrete Mathematics

Question

Prove that for all integers $n$, $n(n + 2)(n + 4)$ is divisible by 3.

Solution

There are only 3 possible remainders when dividing by 3, namely 0, 1, 2.

If $n$ has the remainder 0 (that is, $n \mod 3 = 0$), it means that $n$ is divisible by 3 and hence $n(n + 2)(n + 4)$ is divisible by 3 too because $n$ is a multiplier of the expression $n(n + 2)(n + 4)$.

If $n$ has the remainder 1 (that is, $n \mod 3 = 1$), then $(n + 2)$ is divisible by 3 because $(n + 2)$ has the remainder $1 + 2 = 3$ and it is the same as to have the remainder $0 \mod 3$, therefore $n(n + 2)(n + 4)$ is divisible by 3 too.

If $n$ has the remainder 2 (that is, $n \mod 3 = 2$), then $(n + 4)$ is divisible by 3 because $(n + 4)$ has the remainder $2 + 4 = 6$ and it is the same as to have the remainder $0 \mod 3$, therefore $n(n + 2)(n + 4)$ is divisible by 3 too.

Thus, in any case $n(n + 2)(n + 4)$ is divisible by 3.

Answer provided by https://www.AssignmentExpert.com