Find the units digit of ## 3^{100} ## by the use of Fermat's theorem

[Math100]

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 $1$.

 

[fresh_42]

Well written!