To find the unit digit of the expression \(3^{68} \times 4^{73} \times 15^{101}\), we need to determine the unit digit of each factor separately and then multiply them.
Unit Digit of Powers
We analyze the unit digit of each term:
Finding Unit Digit of \(3^{68}\)
- The unit digits of powers of 3 follow a repeating cycle: 3, 9, 7, 1. The cycle length is 4.
- To find the unit digit of \(3^{68}\), we look at the exponent 68. We find the remainder when 68 is divided by the cycle length, 4.
- Calculation: \(68 \div 4 = 17\) with a remainder of 0.
- When the remainder is 0, the unit digit is the last digit in the cycle, which is 1.
Finding Unit Digit of \(4^{73}\)
- The unit digits of powers of 4 follow a repeating cycle: 4, 6. The cycle length is 2.
- To find the unit digit of \(4^{73}\), we look at the exponent 73. We find the remainder when 73 is divided by the cycle length, 2.
- Calculation: \(73 \div 2 = 36\) with a remainder of 1.
- When the remainder is 1, the unit digit is the first digit in the cycle, which is 4.
Finding Unit Digit of \(15^{101}\)
- Any positive integer power of a number ending in 5 will always have a unit digit of 5.
- Therefore, the unit digit of \(15^{101}\) is 5.
Calculate Final Unit Digit
Now, we multiply the unit digits of each factor:
- Unit digit of \(3^{68}\) is 1.
- Unit digit of \(4^{73}\) is 4.
- Unit digit of \(15^{101}\) is 5.
The unit digit of the product \(3^{68} \times 4^{73} \times 15^{101}\) is the unit digit of the product of their unit digits:
Unit digit of \((1 \times 4 \times 5)\)
Calculation: \(1 \times 4 \times 5 = 20\).
The unit digit of 20 is 0.
Important Questions from Unit Digit
The unit digit in 4 × 38 × 764 × 1256 is:
How many times does the number 3 occur in unit's place for numbers ranging from 1 to 100?
A. 20
B. 11
C. 10
D. 19
Find the unit digit in the given factor (3451) 51 × (531) 43 .
A. 6
B. 4
C. 1
D. 9
What is the unit digit of 178 × 593 + 157?
Find the unit digit of $(123)^{123} \times (347)^{347} \times (568)^{568}$.
