- #1

Math100

- 756

- 204

- Homework Statement
- Prove the assertion below:

If the integer ## a ## is not divisible by ## 2 ## or ## 3 ##, then ## a^{2}\equiv 1\pmod {24} ##.

- Relevant Equations
- None.

Proof:

Suppose that the integer ## a ## is not divisible by ## 2 ## or ## 3 ##.

Then ## a\equiv 1, 5, 7, 11, 13, 17, 19 ## or ## 23\pmod {24} ##.

Note that ## a\equiv b\pmod {n}\implies a^{2}\equiv b^{2}\pmod {n} ##.

Thus ## a^{2}\equiv 1, 25, 49, 121, 169, 289, 361 ## or ## 529\pmod {24}\implies a^{2}\equiv 1\pmod {24} ##.

Therefore, if the integer ## a ## is not divisible by ## 2 ## or ## 3 ##, then ## a^{2}\equiv 1\pmod {24} ##.

Suppose that the integer ## a ## is not divisible by ## 2 ## or ## 3 ##.

Then ## a\equiv 1, 5, 7, 11, 13, 17, 19 ## or ## 23\pmod {24} ##.

Note that ## a\equiv b\pmod {n}\implies a^{2}\equiv b^{2}\pmod {n} ##.

Thus ## a^{2}\equiv 1, 25, 49, 121, 169, 289, 361 ## or ## 529\pmod {24}\implies a^{2}\equiv 1\pmod {24} ##.

Therefore, if the integer ## a ## is not divisible by ## 2 ## or ## 3 ##, then ## a^{2}\equiv 1\pmod {24} ##.