- #1
Math100
- 802
- 222
- Homework Statement
- Find the units digit of ## 3^{100} ## by the use of Fermat's theorem.
- Relevant Equations
- None.
Consider modulo ## 10 ##.
Then ## 10=5\cdot 2 ##.
Applying the Fermat's theorem produces: ## 3^{4}\equiv 1\pmod {5} ##.
This means ## (3^{4})^{25}=3^{100}\equiv 1\pmod {5} ##.
Observe that ## 3\equiv 1\pmod {2}\implies 3^{100}\equiv 1\pmod {2} ##.
Now we have ## 5\mid (3^{100}-1) ## and ## 2\mid (3^{100}-1) ##.
Thus ## (5\cdot 2)\mid (3^{100}-1)\implies 3^{100}\equiv 1\pmod {10} ##.
Therefore, the units digit of ## 3^{100} ## is ## 1 ##.
Then ## 10=5\cdot 2 ##.
Applying the Fermat's theorem produces: ## 3^{4}\equiv 1\pmod {5} ##.
This means ## (3^{4})^{25}=3^{100}\equiv 1\pmod {5} ##.
Observe that ## 3\equiv 1\pmod {2}\implies 3^{100}\equiv 1\pmod {2} ##.
Now we have ## 5\mid (3^{100}-1) ## and ## 2\mid (3^{100}-1) ##.
Thus ## (5\cdot 2)\mid (3^{100}-1)\implies 3^{100}\equiv 1\pmod {10} ##.
Therefore, the units digit of ## 3^{100} ## is ## 1 ##.