To find the unit digit of \( 9^{1234} \), we can look for a pattern in the units digit of powers of 9.

Let’s compute the first few powers of 9 and observe the units digits:

\[
9^1 = 9 \quad \text{(units digit is 9)}
\]
\[
9^2 = 81 \quad \text{(units digit is 1)}
\]
\[
9^3 = 729 \quad \text{(units digit is 9)}
\]
\[
9^4 = 6561 \quad \text{(units digit is 1)}
\]

The units digits repeat in a cycle of 2: \( 9, 1 \).

Now, we divide the exponent \( 1234 \) by the cycle length (which is 2):
\[
1234 \div 2 = 617 \text{ remainder } 0
\]
Since the remainder is 0, the units digit of \( 9^{1234} \) corresponds to the units digit of \( 9^2 \), which is **1**.

Thus, the unit digit of \( 9^{1234} \) is **1**.

Do you want further details or have any questions?

Here are 5 related questions for further practice:
1. What is the unit digit of \( 7^{345} \)?
2. Find the unit digit of \( 2^{100} \).
3. What is the unit digit of \( 5^{1000} \)?
4. What is the unit digit of \( 6^{789} \)?
5. Find the unit digit of \( 3^{4567} \).

**Tip:** Recognizing patterns in the unit digits of powers can significantly speed up solving large exponent problems!

Discover the solution to finding the unit digit of 9^1234 by recognizing the cyclic nature of unit digits in powers. Learn how to apply number theory concepts to solve this large exponent problem. Suitable for Grades 8-10 students.

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