How to Find the Unit Digit of a Number: A Step-by-Step Guide

How to Find the Unit Digit of a Number: A Step-by-Step Guide

How to Find the Unit Digit of a Number

Understanding how to find the unit digit of a number is a vital concept in mathematics, especially when dealing with large numbers or exponents. Whether you’re tackling competitive exams or solving mathematical puzzles, knowing how to efficiently calculate unit digits is crucial.

In this article, we’ll explain how to find the unit digit of any given number using simple techniques, including remainder division and cyclicity patterns.

What is the Unit Digit?

The unit digit is the rightmost digit of any given number, often referred to as the “ones place” or ones digit. For example, in the number 3451, the unit digit is 1.


How to Find the Unit Digit of a Number

The unit digit can be found using two primary methods: visual identification and calculation through patterns. Let’s explore each one.

1. Visual Identification of Unit Digit

For most numbers, the unit digit can be easily identified by simply looking at the number and focusing on the rightmost digit before the decimal point.

2. Cyclicity or Pattern of Unit Digits for Exponents

In some cases, especially when dealing with exponentiation, the unit digit follows a specific pattern or cycle. For instance, consider powers of 2:

  • 2^1 = 2
  • 2^2 = 4
  • 2^3 = 8
  • 2^4 = 16
  • 2^5 = 32 (and so on)

This repeating cycle of 2, 4, 8, 6 helps us determine the unit digit for any exponent of 2.

Pro Tip: Identify the remainder when dividing the exponent by the length of the cycle. For example, to find the unit digit of 2^60, divide 60 by 4 (the length of the cycle). The remainder is 0, so the unit digit of 2^60 is 6.


Understanding Remainder and Modulo 10

Another efficient method to find the unit digit is by finding the remainder when a number is divided by 10. This approach is straightforward and works well with large numbers.

For example:

  • 3456 ÷ 10 gives a remainder of 6, so the unit digit of 3456 is 6.

Solved Examples: Calculating Unit Digits

Let’s look at some practical examples to better understand how to calculate the unit digit.

Example 1: Unit Digit of a Product

Problem: Find the unit digit of the product 684 × 759 × 413 × 676.

Solution: The unit digit of the product is determined by multiplying the unit digits of each number:

  • 684 → Unit digit = 4
  • 759 → Unit digit = 9
  • 413 → Unit digit = 3
  • 676 → Unit digit = 6

Now, calculate the product of the unit digits:
4 × 9 × 3 × 6 = 648
The unit digit of 648 is 8, so the unit digit of the entire product is 8.


Example 2: Unit Digit of Large Powers

Problem: Find the unit digit of (3547)^153 × (251)^72.

Solution:

  • The unit digit of 3547 is 7.
  • The unit digit of 251 is 1.

Now, using the pattern for powers of 7, we know the cycle is 7, 9, 3, 1.

  • Since 153 ÷ 4 leaves a remainder of 1, the unit digit of 7^153 is 7.
  • For 251^72, the unit digit of any power of 1 is always 1.

Hence, the unit digit of the product is 7 × 1 = 7.


Final Tips for Finding Unit Digits

  • For products: Always multiply the unit digits first and then find the unit digit of the resulting product.
  • For large exponents: Recognize cycles or use modulo division to reduce the problem size.
  • For sums: Use the unit digits of the individual terms and apply the same principles.

Conclusion

By mastering these methods, you’ll be able to quickly and accurately find the unit digit of any number, whether it’s a large product, a sum, or an exponentiation.

How to Find the Unit Digit of a Number | Step-by-Step Guide
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