To find the unit digit of \( 7^{95} - 3^{58} \), we can work with the properties of numbers modulo 10. This helps us identify patterns in the units digits of powers of integers. Let's break this down:

### Step 1: Find the unit digit of \( 7^{95} \)
We first observe the pattern of the units digits of powers of 7:

\[
\begin{aligned}
7^1 & = 7 \quad \text{(units digit: 7)} \\
7^2 & = 49 \quad \text{(units digit: 9)} \\
7^3 & = 343 \quad \text{(units digit: 3)} \\
7^4 & = 2401 \quad \text{(units digit: 1)} \\
\end{aligned}
\]

The units digits repeat every 4 powers: \( 7, 9, 3, 1 \).

To find \( 7^{95} \mod 10 \), we divide 95 by 4 and focus on the remainder:

\[
95 \div 4 = 23 \text{ remainder } 3
\]

So, the units digit of \( 7^{95} \) is the same as that of \( 7^3 \), which is 3.

### Step 2: Find the unit digit of \( 3^{58} \)
Similarly, we look for the pattern in the units digits of powers of 3:

\[
\begin{aligned}
3^1 & = 3 \quad \text{(units digit: 3)} \\
3^2 & = 9 \quad \text{(units digit: 9)} \\
3^3 & = 27 \quad \text{(units digit: 7)} \\
3^4 & = 81 \quad \text{(units digit: 1)} \\
\end{aligned}
\]

The units digits also repeat every 4 powers: \( 3, 9, 7, 1 \).

To find \( 3^{58} \mod 10 \), we divide 58 by 4 and focus on the remainder:

\[
58 \div 4 = 14 \text{ remainder } 2
\]

So, the units digit of \( 3^{58} \) is the same as that of \( 3^2 \), which is 9.

### Step 3: Calculate the units digit of \( 7^{95} - 3^{58} \)
Now, we subtract the units digits:

\[
7^{95} - 3^{58} \equiv 3 - 9 \mod 10 = -6 \mod 10 = 4
\]

Thus, the units digit of \( 7^{95} - 3^{58} \) is **4**.

Would you like further details or clarifications?

### Related Questions:
1. What is the units digit of \( 9^{125} \)?
2. How do we find the units digit of \( 5^{100} \)?
3. What is the cyclic pattern of units digits for powers of 6?
4. How does mod arithmetic help in simplifying power-related problems?
5. Can the units digit of \( a^b \) ever exceed 9?

### Tip:
Always look for patterns in the units digits of powers, as they often repeat in cycles, simplifying the problem.

Explore how to find the unit digit of 7^95 - 3^58 using modular arithmetic and cyclic patterns. Understand the step-by-step breakdown of how to solve this number theory problem. Suitable for students in Grades 9-12.

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