What is the unit digit in 189^723? A] 7 B] 9 C] 1 D] 3 Options : A B C D

1. Break Down the Problem

To find the unit digit of $189^{723}$, we only need to consider the unit digit of the base number, which is $9$ in this case. Therefore, we will find the unit digit of $9^{723}$.

2. Relevant Concepts

The unit digits of the powers of $9$ follow a pattern:

• $9^1$ has a unit digit of $9$
• $9^2$ has a unit digit of $1$
• The pattern repeats every two powers: $9, 1, 9, 1, \ldots$

3. Analysis and Detail

To determine the unit digit of $9^{723}$, we find the exponent $723$ modulo $2$: $723 \mod 2 = 1$ Since $723$ is odd, the unit digit corresponds to the first element in the pattern which is $9$.

4. Verify and Summarize

Through our analysis, we see that the unit digit of $9^1$ is $9$. Since $723$ is odd, $9^{723}$ also has a unit digit of $9$.

Final Answer

The unit digit in $189^{723}$ is 9 (Option B).