Question

The digit in the unit’s place of the product $3^{999} \times 7^{1000}$ is __________.

• A. 7
• B. 1
• C. 3
• D. 9

Hint:

For unit digit problems, always check the cyclicity of powers (mod 4 for numbers ending with 3, 7, 9, 2). Multiply only the last digits.

Answer 1

The Correct Option is B

Step 1: Find the unit digit of $3^{999$.}
The powers of $3$ follow a cycle of unit digits: $3, 9, 7, 1$. This repeats every 4 terms. Now, $999 \div 4$ leaves a remainder of $3$. Hence, $3^{999}$ will have the same unit digit as $3^3$, which is $7$.
Step 2: Find the unit digit of $7^{1000$.}
The powers of $7$ follow a cycle of unit digits: $7, 9, 3, 1$. This repeats every 4 terms. Now, $1000 \div 4$ leaves a remainder of $0$. Hence, $7^{1000}$ will have the same unit digit as $7^4$, which is $1$.
Step 3: Multiply the unit digits.
The unit digits are $7 \times 1 = 7$.
Step 4: Verify carefully.
Wait — but let us double-check: - $3^{999}$ ends in $7$. - $7^{1000}$ ends in $1$. So, product’s unit digit $= 7 \times 1 = 7$. Therefore, the correct unit digit is $7$, not $1$. Final Answer: \[ \boxed{7} \]