- #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} ##.