- #1

Math100

- 779

- 220

- Homework Statement
- Prove the following statement:

Any one of the integers ## 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ## can occur as the units digit of ## a^{3} ##.

- 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} ##.

Thus ## a^{3}\equiv 0, 1, 8, 27, 64, 125, 216, 343, 512 ##, or ## 729\pmod {10} ##.

Therefore, anyone of the integers ## 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ## can occur as the units digit of ## a^{3} ##.

Let ## a ## be any integer.

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

Thus ## a^{3}\equiv 0, 1, 8, 27, 64, 125, 216, 343, 512 ##, or ## 729\pmod {10} ##.

Therefore, anyone of the integers ## 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ## can occur as the units digit of ## a^{3} ##.