Unit digits concept
UNIT DIGITS
APTITUDE
Unit digits definition:
The unit digit of a number is the last digit in the number, which represents the "one's place." It is the digit that remains when all higher place values (like tens, hundreds, etc.) are removed.
Examples:
1. The unit digit of 345 is 5.
2. The unit digit of 78 is 8.
3. The unit digit of 1024 is 4.
The unit digit is often used in problems involving:
● Multiplications (to find the unit digit of a product).
● Powers (to determine repeating patterns of unit digits).
● Modular arithmetic (in competitive exams for quick calculations).
SOLVING METHODS
Method 1: Direct Observation of Unit Digits
● The unit digit of a number follows a repeating cycle based on its powers.
● Example cycles:
○ 0n=00^n = 0
○ 1n=11^n = 1
○ 5n=55^n = 5
○ 6n=66^n = 6
● For other numbers, the unit digits follow a cyclic pattern.
Method 2:Identifying Odd and Even Powers
● The last digit of 4 cycles between 4 and 6:
○ If the exponent is odd → unit digit is 4.
○ If the exponent is even → unit digit is 6.
● The last digit of 9 cycles between 9 and 1:
○ If the exponent is odd → unit digit is 9.
○ If the exponent is even → unit digit is 1.
Method 3: Checking Power Cycles for 2, 3, 7, 8
● When the last digit is 2, 3, 7, or 8, the power follows a cycle of 4.
● To find the unit digit:
1. Divide the exponent by 4 and get the remainder. 2. Use the remainder to determine the unit digit from the cycle.
EXAMPLES
Method 1: Direct Observation of Unit Digits
● This method is used when the unit digit follows a fixed pattern without needing further calculation.
Example 1
Find the unit digit of 7355735^57355.
● The unit digit of 735 is 5.
● From the cycle: 5n=55^n = 55n=5 for any power.
● Final unit digit: 5.
Method 2: Identifying Odd and Even Powers
● This method is used for numbers like 4 and 9, which have alternating unit digits based on even or odd powers.
Example 2
Find the unit digit of 1948194^81948 and 1949194^91949.
● The unit digit of 194 is 4.
● 4 has a cycle:
○ Odd power → 4
○ Even power → 6
● Since 8 is even, unit digit = 6.
● Since 9 is odd, unit digit = 4.
Method 3: Checking Power Cycles for 2, 3, 7, 8 ● This method is used when the last digit is 2, 3, 7, or 8.
Example 3: Find the unit digit of 374537^{45}3745.
● The unit digit of 37 is 7.
● The cycle of 7 is: 7 → 9 → 3 → 1 (Repeats every 4 powers).
● Divide exponent by 4:
○ 45 mod 4=145 \mod 4 = 145mod4=1.
● The remainder is 1, so the unit digit is the first in the cycle, which is 7.
