Question

What is the unit digit of $(234)^{100} + (234)^{101}$?

• A. 6
• B. 2
• C. 4
• D. 0

Hint:

For unit digit problems, ignore all digits except the last one and use cyclic patterns of powers to simplify calculations.

Answer 1

The Correct Option is D

Step 1: Observe that only the unit digit matters.
To find the unit digit of large powers, we only need the unit digit of the base number.
The unit digit of 234 is 4.
Step 2: Rewrite the expression using unit digits.
\[ (234)^{100} + (234)^{101} \]
becomes
\[ 4^{100} + 4^{101} \]
Step 3: Find the pattern of powers of 4.
\[ 4^1 = 4 \]
\[ 4^2 = 16 \Rightarrow \text{unit digit } = 6 \]
\[ 4^3 = 64 \Rightarrow \text{unit digit } = 4 \]
\[ 4^4 = 256 \Rightarrow \text{unit digit } = 6 \]
Thus, the unit digits repeat in a cycle of 2:
\[ 4, 6, 4, 6, \dots \]
Step 4: Determine the unit digit of $4^{100$.}
Since 100 is even,
\[ 4^{100} \text{ has unit digit } 6 \]
Step 5: Determine the unit digit of $4^{101$.}
Since 101 is odd,
\[ 4^{101} \text{ has unit digit } 4 \]
Step 6: Add the unit digits.
\[ 6 + 4 = 10 \]
Step 7: Find the final unit digit.
The unit digit of 10 is 0.
Step 8: Final conclusion.
Hence, the unit digit of $(234)^{100} + (234)^{101}$ is 0.