Prove that ## a^{3}\equiv 0, 1 ##, or ## 6\pmod {7} ##?

[Math100]

Homework Statement

Prove the assertions below:
For any integer $a$, $a^{3}\equiv 0, 1,$ or $6\pmod {7}$.

Relevant Equations

None.

Proof:

Let $a$ be any integer.
Then $a\equiv 0, 1, 2, 3, 4, 5,$ or $6\pmod {7}$.
Note that $a\equiv b\pmod {n}\implies a^{3}\equiv b^3\pmod{n}$.
This means $a^{3}\equiv 0, 1, 8, 27, 64, 125$ or $216\pmod{7}\implies a^{3}\equiv 0, 1, 1, 6, 1, 6$ or $6\pmod {7}$.
Thus $a^{3}\equiv 0, 1$ or $6\pmod {7}$.
Therefore, $a^{3}\equiv 0, 1,$ or $6\pmod {7}$ for any integer $a$.

 

[fresh_42]

Homework Statement:: Prove the assertions below:
For any integer $a$, $a^{3}\equiv 0, 1,$ or $6\pmod {7}$.
Relevant Equations:: None.

Proof:

Let $a$ be any integer.
Then $a\equiv 0, 1, 2, 3, 4, 5,$ or $6\pmod {7}$.
Note that $a\equiv b\pmod {n}\implies a^{3}\equiv b^3\pmod{n}$.
This means $a^{3}\equiv 0, 1, 8, 27, 64, 125$ or $216\pmod{7}\implies a^{3}\equiv 0, 1, 1, 6, 1, 6$ or $6\pmod {7}$.
Thus $a^{3}\equiv 0, 1$ or $6\pmod {7}$.
Therefore, $a^{3}\equiv 0, 1,$ or $6\pmod {7}$ for any integer $a$.

Correct.