Prove that ## a^{3}+1 ## is divisible by ## 7 ##.

[Math100]

Homework Statement

If $7\nmid a$, prove that either $a^{3}+1$ or $a^{3}-1$ is divisible by $7$.

Relevant Equations

None.

Proof:

Suppose $7\nmid a$.
Applying the Fermat's theorem produces:
$a^{7-1}\equiv 1\pmod {7}\implies a^{6}\equiv 1\pmod {7}$.
This means $7\mid (a^{6}-1)$.
Observe that $a^{6}-1=(a^{3}-1)(a^{3}+1)$.
Thus $7\nmid (a^{3}-1)\implies 7\mid (a^{3}+1)$ and $7\nmid (a^{3}+1)\implies 7\mid (a^{3}-1)$.
Therefore, if $7\nmid a$, then either $a^{3}+1$ or $a^{3}-1$ is divisible by $7$.

 

[fresh_42]

Correct.

You basically use that $7$ is a prime number. That means, $7\neq \pm1$ and if $7\,|\,a\cdot b \Longrightarrow 7\,|\,a \text{ or } 7\,|\,b.$ This is the correct definition of a prime number.