The digit in the U it's place of the number represent by (7^95-3^58) is

To find the digit in the units place of the number represented by \(7^{95} - 3^{58}\), we need to determine the units digit of both \(7^{95}\) and \(3^{58}\) individually, and then subtract the second from the first.

### Step 1: Determine the units digit of \(7^{95}\)
The units digits of powers of 7 follow a cycle:
\[
\begin{aligned}
7^1 & = 7 &\text{(units digit = 7)} \\
7^2 & = 49 &\text{(units digit = 9)} \\
7^3 & = 343 &\text{(units digit = 3)} \\
7^4 & = 2401 &\text{(units digit = 1)} \\
\end{aligned}
\]
So the cycle of the units digits is \(7, 9, 3, 1\), repeating every 4 terms.

To find the units digit of \(7^{95}\), we calculate the remainder when 95 is divided by 4:
\[
95 \div 4 = 23 \text{ remainder } 3
\]
Thus, the units digit of \(7^{95}\) is the same as the units digit of \(7^3\), which is 3.

### Step 2: Determine the units digit of \(3^{58}\)
The units digits of powers of 3 follow a cycle:
\[
\begin{aligned}
3^1 & = 3 &\text{(units digit = 3)} \\
3^2 & = 9 &\text{(units digit = 9)} \\
3^3 & = 27 &\text{(units digit = 7)} \\
3^4 & = 81 &\text{(units digit = 1)} \\
\end{aligned}
\]
So the cycle of the units digits is \(3, 9, 7, 1\), repeating every 4 terms.

To find the units digit of \(3^{58}\), we calculate the remainder when 58 is divided by 4:
\[
58 \div 4 = 14 \text{ remainder } 2
\]
Thus, the units digit of \(3^{58}\) is the same as the units digit of \(3^2\), which is 9.

### Step 3: Subtract the units digits
Now, subtract the units digit of \(3^{58}\) from the units digit of \(7^{95}\):
\[
3 - 9 = -6
\]
Since we want the units digit, we take this modulo 10:
\[
-6 \equiv 4 \pmod{10}
\]
So, the units digit of \(7^{95} - 3^{58}\) is **4**.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you find the units digit of large exponents efficiently?
2. What is the significance of the modulo operation in finding units digits?
3. Can you calculate the last two digits of \(7^{95} - 3^{58}\)?
4. How do units digits cycles differ between different numbers?
5. What are the properties of modular arithmetic in the context of large numbers?

**Tip:** Understanding patterns in units digits for powers is a powerful tool in solving many exponent-related problems!

Learn how to find the units digit of the number represented by 7 to the power of 95 minus 3 to the power of 58. This problem explores the cycles of units digits in powers of 7 and 3, and uses modular arithmetic to determine the final digit. Suitable for high school students interested in number theory and modular arithmetic.

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