- #1

Math100

- 782

- 220

- 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 ##.

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 ##.