- #1

Math100

- 768

- 207

- Homework Statement
- Prove the assertion below:

For any integer ## a ##, ## a^{4}\equiv 0 ## or ## 1\pmod {5} ##.

- Relevant Equations
- None.

Proof:

Let ## a ## be any integer.

Then ## a\equiv 0, 1, 2, 3 ## or ## 4\pmod {5} ##.

Note that ## a\equiv b\pmod {n}\implies a^{4}\equiv b^{4}\pmod {n} ##.

This means ## a^{4}\equiv 0, 1, 16, 81 ## or ## 256\pmod {5}\implies a^{4}\equiv 0, 1, 1, 1 ## or ## 1\pmod {5} ##.

Thus ## a^{4}\equiv 0 ## or ## 1\pmod {5} ##.

Therefore, ## a^{4}\equiv 0 ## or ## 1\pmod {5} ## for any integer ## a ##.

Let ## a ## be any integer.

Then ## a\equiv 0, 1, 2, 3 ## or ## 4\pmod {5} ##.

Note that ## a\equiv b\pmod {n}\implies a^{4}\equiv b^{4}\pmod {n} ##.

This means ## a^{4}\equiv 0, 1, 16, 81 ## or ## 256\pmod {5}\implies a^{4}\equiv 0, 1, 1, 1 ## or ## 1\pmod {5} ##.

Thus ## a^{4}\equiv 0 ## or ## 1\pmod {5} ##.

Therefore, ## a^{4}\equiv 0 ## or ## 1\pmod {5} ## for any integer ## a ##.