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To solve this problem, we need to find the units digit of the product \( 3^{999} \times 7^{1000} \).

### Step 1: Analyze the units digit of \( 3^{999} \)
The units digits of powers of 3 follow a repeating cycle of 4:
\[
\begin{aligned}
3^1 & = 3 \\
3^2 & = 9 \\
3^3 & = 27 \quad (\text{units digit is } 7) \\
3^4 & = 81 \quad (\text{units digit is } 1)
\end{aligned}
\]
Thus, the cycle of units digits for powers of 3 is: \( 3, 9, 7, 1 \).

To find the units digit of \( 3^{999} \), we divide 999 by 4:
\[
999 \div 4 = 249 \text{ remainder } 3
\]
Since the remainder is 3, the units digit of \( 3^{999} \) corresponds to the units digit of \( 3^3 \), which is 7.

### Step 2: Analyze the units digit of \( 7^{1000} \)
The units digits of powers of 7 also follow a repeating cycle of 4:
\[
\begin{aligned}
7^1 & = 7 \\
7^2 & = 49 \quad (\text{units digit is } 9) \\
7^3 & = 343 \quad (\text{units digit is } 3) \\
7^4 & = 2401 \quad (\text{units digit is } 1)
\end{aligned}
\]
Thus, the cycle of units digits for powers of 7 is: \( 7, 9, 3, 1 \).

To find the units digit of \( 7^{1000} \), we divide 1000 by 4:
\[
1000 \div 4 = 250 \text{ remainder } 0
\]
Since the remainder is 0, the units digit of \( 7^{1000} \) corresponds to the units digit of \( 7^4 \), which is 1.

### Step 3: Multiply the units digits
Now, we multiply the units digits of \( 3^{999} \) and \( 7^{1000} \):
\[
7 \times 1 = 7
\]

Thus, the units digit of \( 3^{999} \times 7^{1000} \) is **7**.

### Final Answer:
The answer is \( \boxed{7} \).

Would you like further details or have any other questions?

Here are 5 related questions for further practice:
1. What is the units digit of \( 2^{1234} \times 5^{6789} \)?
2. What is the units digit of \( 4^{1001} \)?
3. Find the units digit of \( 9^{245} \times 6^{379} \).
4. How would you calculate the units digit of \( 8^{9876} \)?
5. What is the units digit of \( 12^{999} \times 13^{1000} \)?

**Tip:** When dealing with large powers, look for repeating cycles in the units digits of base numbers to simplify the calculation!

The digit in the unit's place of the product 3^999 × 7^1000 is ______.

Learn how to determine the unit's digit of the product 3^999 × 7^1000. This problem explores number theory concepts such as cycles in units digits and modular arithmetic, suitable for high school students in Grades 9-12.

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