Question

What is the unit digit in $7^{105}$?

Answer 1

Determine the unit digit in $7^{105}$.

Find the unit digits up to $7^4$.

$7^1=7$. So, the unit digit is $7$.

$7^2=49$. So, the unit digit is $9$.

$7^3=343$. So, the unit digit is $3$.

$7^4=2401$. So, the unit digit is $1$.

Now, writing the $7^{105}$ in term of $7^4$,

we get,

$7^{105}={\left(7^4\right)}^{26}·7\cdots \left(1\right)$

Since the unit digit of $7^4$ is $1$, substitute $1$ into equation $\left(1\right)$.

$\begin{array}{rcl}{7}^{105}& =& {\left(1\right)}^{26}·7\\ & \Rightarrow & 7^{105}=1·7\\ & \Rightarrow & 7^{105}=7\end{array}$

Hence, the unit digit in $7^{105}$ is $7$.