- #1

Math100

- 756

- 204

- Homework Statement
- For any integer ## a ##, show that ## a^{2}-a+7 ## ends in one of the digits ## 3, 7 ##, or ## 9 ##.

- Relevant Equations
- None.

Proof:

Let ## a ## be any integer.

Then ## a\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8 ## or ## 9\pmod {10} ##.

Note that ## a^{2}\equiv 0, 1, 4, 9, 6, 5, 6, 9, 4 ## or ## 1\pmod {10} ##.

Thus ## a^{2}-a+7\equiv 7, 7, 9, 9, 7, 7, 7, 9, 9 ## or ## 7\pmod {10} ##.

Therefore, ## a^{2}-a+7 ## ends in one of the digits ## 3, 7 ##, or ## 9 ## for any integer ## a ##.

Let ## a ## be any integer.

Then ## a\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8 ## or ## 9\pmod {10} ##.

Note that ## a^{2}\equiv 0, 1, 4, 9, 6, 5, 6, 9, 4 ## or ## 1\pmod {10} ##.

Thus ## a^{2}-a+7\equiv 7, 7, 9, 9, 7, 7, 7, 9, 9 ## or ## 7\pmod {10} ##.

Therefore, ## a^{2}-a+7 ## ends in one of the digits ## 3, 7 ##, or ## 9 ## for any integer ## a ##.