Unit Digit - Important Points
Understanding the Unit Digit: An Essential Concept in Number Theory
The unit digit, also known as the ones digit or the last digit, plays a crucial role in understanding the properties and patterns of numbers. It refers to the digit in the one's place of a number and has a significant impact on various mathematical operations. In this write-up, we will explore the concept of the unit digit, its properties, and some essential formulas to calculate it, with the help of suitable examples.
Definition of Unit Digit: The unit digit of a number is the rightmost digit when it is expressed in base 10 (decimal system). For instance, in the number 5472, the unit digit is 2. It is essential to recognize that the unit digit ranges from 0 to 9.
Cyclic Patterns of Unit Digits: One of the fundamental aspects of unit digits is the cyclic pattern they follow when specific powers of a number are calculated. For instance, let's consider the unit digits of the positive powers of 7:
7^1 = 7 (unit digit: 7) 7^2 = 49 (unit digit: 9) 7^3 = 343 (unit digit: 3) 7^4 = 2401 (unit digit: 1)
From the above, we observe that the unit digits of powers of 7 repeat in a cycle: 7, 9, 3, and 1. This cyclic pattern will continue as we calculate higher powers of 7.
Here's a tabular form showing the unit digits of all numbers from 1 to 9 and their cyclical patterns:
|
Number (n) |
n^1 |
n^2 |
n^3 |
n^4 |
n^5 |
n^6 |
n^7 |
n^8 |
n^9 |
|
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
2 |
2 |
4 |
8 |
6 |
2 |
4 |
8 |
6 |
2 |
|
3 |
3 |
9 |
7 |
1 |
3 |
9 |
7 |
1 |
3 |
|
4 |
4 |
6 |
4 |
6 |
4 |
6 |
4 |
6 |
4 |
|
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
|
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
|
7 |
7 |
9 |
3 |
1 |
7 |
9 |
3 |
1 |
7 |
|
8 |
8 |
4 |
2 |
6 |
8 |
4 |
2 |
6 |
8 |
|
9 |
9 |
1 |
9 |
1 |
9 |
1 |
9 |
1 |
9 |
In the above table, each row represents the unit digits of the numbers from 1 to 9 raised to different powers (n^1 to n^9). We can observe the cyclical patterns in the unit digits of each number. These cyclical patterns help us quickly identify the unit digit of any power of a given number. For example, the unit digit of 2^7 is 8, the unit digit of 3^4 is 1, and the unit digit of 5^9 is 5.
Properties of Unit Digits: Several properties govern the unit digits of numbers and their operations. Some notable ones are:
1. Addition and Subtraction: When adding or subtracting numbers, we only need to focus on their unit digits. The sum or difference of two numbers will have the same unit digit as the sum or difference of their unit digits. For example:
72 + 54 = 126 (unit digits: 2 + 4 = 6)
2. Multiplication: To find the unit digit of the product of two numbers, we multiply their unit digits. For instance:
7 * 9 = 63 (unit digits: 7 * 9 = 63)
3. Division: When dividing numbers, the unit digit of the quotient depends on the unit digit of the dividend and divisor. However, it is important to consider other digits as well for the correct result.
Important Formulas for Unit Digits
1. Unit Digit of Powers: As seen in the cyclic pattern example for 7, to find the unit digit of any power of a number, determine the remainder when the exponent is divided by the cycle length (in this case, 4). Then, the unit digit will be the digit in that position in the cycle.
2. Unit Digit of Multiplication: To find the unit digit of the product of multiple numbers, calculate the product of their unit digits. The unit digit of the result will be the unit digit of this product.
3. Unit Digit of Division: To find the unit digit of a quotient, consider the unit digit of the dividend and divisor. Calculate their quotient, and the unit digit of this quotient will be the unit digit of the result.
Examples:
-
Find the unit digit of 3^15. Here, the cycle length of 3 is 4 (3, 9, 7, 1). The remainder when 15 is divided by 4 is 3. Thus, the unit digit of 3^15 is the third element in the cycle, which is 7.
-
Determine the last digit of 785 * 432 * 619. To find the last digit of the product, multiply the last digits of each number: 5 * 2 * 9 = 90. The unit digit of 90 is 0.
Understanding the unit digit concept is essential for various mathematical calculations and problem-solving, especially in competitive exams. It helps simplify complex computations and reveals recurring patterns in numbers. By applying the formulas and strategies mentioned above, one can confidently handle questions related to unit digits and enhance their overall numerical abilities.