Math Problem Statement
The digit in the U it's place of the number represent by (7^95-3^58) is
Solution
To find the digit in the units place of the number represented by , we need to determine the units digit of both and individually, and then subtract the second from the first.
Step 1: Determine the units digit of
The units digits of powers of 7 follow a cycle:
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!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponentiation
Modular Arithmetic
Units Digits
Formulas
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Theorems
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Suitable Grade Level
High School