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 7953587^{95} - 3^{58}, we need to determine the units digit of both 7957^{95} and 3583^{58} individually, and then subtract the second from the first.

Step 1: Determine the units digit of 7957^{95}

The units digits of powers of 7 follow a cycle:

71=7(units digit = 7)72=49(units digit = 9)73=343(units digit = 3)74=2401(units digit = 1)\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 7,9,3,17, 9, 3, 1, repeating every 4 terms.

To find the units digit of 7957^{95}, we calculate the remainder when 95 is divided by 4:

95÷4=23 remainder 395 \div 4 = 23 \text{ remainder } 3

Thus, the units digit of 7957^{95} is the same as the units digit of 737^3, which is 3.

Step 2: Determine the units digit of 3583^{58}

The units digits of powers of 3 follow a cycle:

31=3(units digit = 3)32=9(units digit = 9)33=27(units digit = 7)34=81(units digit = 1)\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 3,9,7,13, 9, 7, 1, repeating every 4 terms.

To find the units digit of 3583^{58}, we calculate the remainder when 58 is divided by 4:

58÷4=14 remainder 258 \div 4 = 14 \text{ remainder } 2

Thus, the units digit of 3583^{58} is the same as the units digit of 323^2, which is 9.

Step 3: Subtract the units digits

Now, subtract the units digit of 3583^{58} from the units digit of 7957^{95}:

39=63 - 9 = -6

Since we want the units digit, we take this modulo 10:

64(mod10)-6 \equiv 4 \pmod{10}

So, the units digit of 7953587^{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 7953587^{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!

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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