How To Find The Last Digit Of An Expression: Easy Guide

by SLV Team 56 views

Hey guys! Ever wondered how to quickly find the last digit of a complex mathematical expression? It might seem daunting, but it's actually pretty straightforward once you get the hang of it. In this guide, we'll break down the process step-by-step and tackle some examples together. So, grab your thinking caps, and let's dive in!

Understanding the Basics

Before we jump into solving expressions, let's make sure we're all on the same page with the basic concept. The last digit of a number is simply the digit in the ones place. For example, the last digit of 1245 is 5, and the last digit of 5647 is 7. When we're dealing with addition, subtraction, multiplication, or even exponentiation, there are some neat tricks we can use to find the last digit without having to calculate the entire result.

The key idea here is that the last digit of a sum, difference, or product is determined only by the last digits of the numbers being added, subtracted, or multiplied. This might sound a bit abstract, so let's illustrate it with an example. Think about adding 23 and 45. The last digit of 23 is 3, and the last digit of 45 is 5. When we add these two numbers, we get 68, and the last digit of the result is 8. Notice that 3 + 5 = 8. This isn't a coincidence; it's a fundamental property that we can use to simplify our calculations.

Why Does This Work?

You might be wondering why we can focus solely on the last digits. The reason lies in the way our number system (base 10) works. When you add two numbers, the ones digits are added together first. If the sum is 10 or greater, we carry over to the tens place, but the ones digit of the sum remains determined by the original ones digits. The same principle applies to subtraction and multiplication, though the details of how the digits interact might be slightly different.

For example, when multiplying two numbers, the last digit of the product is determined by the product of the last digits of the original numbers. If you multiply 13 by 17, the last digits are 3 and 7, and their product is 21. The last digit of 21 is 1, which is also the last digit of 13 * 17 = 221. This pattern holds true because the tens, hundreds, and higher-order digits in the original numbers don't contribute to the ones digit of the final result.

Now that we've got the basic concept down, let's apply this knowledge to some actual expressions. We'll start with simple addition and subtraction problems and then move on to more complex scenarios. Remember, the goal is to find the last digit without doing the full calculation. By focusing on the last digits, we can save time and mental effort, which is especially useful in exams or situations where quick calculations are needed.

Addition and Subtraction

Let's start with some addition and subtraction problems. The principle here is straightforward: focus only on the last digits of the numbers involved. Add or subtract these last digits, and the result will give you the last digit of the entire expression. If the result is a two-digit number, just take the last digit of that result.

Example 1: 1245 + 5647

To find the last digit of this expression, we'll look at the last digits of 1245 and 5647, which are 5 and 7, respectively. Now, we simply add these digits:

5 + 7 = 12

The result is 12, which is a two-digit number. We're only interested in the last digit, so we take the 2. Therefore, the last digit of 1245 + 5647 is 2.

To verify, you can perform the full addition: 1245 + 5647 = 6892. Indeed, the last digit is 2, confirming our method.

Example 2: 145781 + 659874

Here, the last digits are 1 and 4. Adding them gives:

1 + 4 = 5

So, the last digit of 145781 + 659874 is 5.

Again, if you were to do the full addition, you'd find that 145781 + 659874 = 805655, which ends in 5.

Example 3: 455412 - 6542

For subtraction, we follow the same principle. The last digits are 2 and 2. Subtracting them gives:

2 - 2 = 0

Thus, the last digit of 455412 - 6542 is 0.

Checking the full subtraction: 455412 - 6542 = 448870, which ends in 0.

Example 4: 45781147 - 451259

The last digits are 7 and 9. Subtracting them gives:

7 - 9 = -2

Wait a minute! We've got a negative result. What do we do now? Remember that we're working in base 10. When the result of subtracting the last digits is negative, we add 10 to it. So,

-2 + 10 = 8

Therefore, the last digit of 45781147 - 451259 is 8.

Performing the full subtraction: 45781147 - 451259 = 45329888, which ends in 8.

Example 5: 4152547 - 145218

The last digits are 7 and 8. Subtracting them gives:

7 - 8 = -1

Again, we have a negative result. Adding 10 to it gives:

-1 + 10 = 9

So, the last digit of 4152547 - 145218 is 9.

Full subtraction: 4152547 - 145218 = 4007329, which ends in 9.

Example 6: 41526 - 415879

The last digits are 6 and 9. Subtracting them gives:

6 - 9 = -3

Adding 10 to it gives:

-3 + 10 = 7

So, the last digit of 41526 - 415879 is 7.

Full subtraction: 41526 - 415879 = -374353. Since we're only interested in the last digit, we consider the absolute value's last digit, which is 3. However, if we were looking for the last digit of the result (including the sign), we would need to consider the full calculation. In this case, the last digit of -374353 is 3, but the question typically refers to the last digit of the magnitude.

Key Takeaways for Addition and Subtraction

  • Focus on the last digits of the numbers.
  • Add or subtract the last digits.
  • If the result is a two-digit number, take its last digit.
  • If the result of subtraction is negative, add 10 to it.

With these principles in mind, finding the last digit of addition and subtraction expressions becomes much simpler and faster. Now, let's move on to more complex operations like multiplication and exponentiation, where the same basic ideas can be extended with a few extra tricks.

Multiplication

Multiplication is where things get a little more interesting, but the core principle remains the same: we only need to focus on the last digits. The last digit of the product of two numbers is determined solely by the product of their last digits. If the product of the last digits is a two-digit number, we take its last digit as the final answer.

The Rule of Last Digits in Multiplication

When multiplying two numbers, the last digit of the result is the last digit of the product of their last digits. This rule stems from the distributive property of multiplication and the nature of our base-10 number system. Higher-order digits (tens, hundreds, etc.) don't influence the ones digit in the final product, making our task much simpler.

Example: Let's consider 23 * 47

  • Last digit of 23 is 3
  • Last digit of 47 is 7

Multiply the last digits: 3 * 7 = 21

The last digit of the result (21) is 1. Therefore, the last digit of 23 * 47 is 1. If you calculate 23 * 47, you get 1081, which indeed ends in 1.

Breaking Down the Process with More Examples

To solidify this concept, let’s look at a series of examples with detailed explanations.

Example 1: Find the last digit of 15 * 26

  1. Identify the last digits: 5 and 6
  2. Multiply the last digits: 5 * 6 = 30
  3. The last digit of 30 is 0. Therefore, the last digit of 15 * 26 is 0.

Example 2: Determine the last digit of 124 * 35

  1. Identify the last digits: 4 and 5
  2. Multiply the last digits: 4 * 5 = 20
  3. The last digit of 20 is 0. Thus, the last digit of 124 * 35 is 0.

Example 3: What is the last digit of 78 * 92?

  1. Last digits: 8 and 2
  2. Multiply them: 8 * 2 = 16
  3. The last digit of 16 is 6. So, the last digit of 78 * 92 is 6.

Example 4: Find the last digit of 123 * 456

  1. Last digits: 3 and 6
  2. Multiply them: 3 * 6 = 18
  3. The last digit of 18 is 8. Therefore, the last digit of 123 * 456 is 8.

Example 5: Calculate the last digit of 999 * 888

  1. Last digits: 9 and 8
  2. Multiply them: 9 * 8 = 72
  3. The last digit of 72 is 2. So, the last digit of 999 * 888 is 2.

Key Observations and Patterns

As we go through these examples, we can notice some interesting patterns. For instance, any multiplication involving a number ending in 5 and an even number will always result in a product ending in 0. Similarly, multiplying any number by a number ending in 0 will always result in a product ending in 0.

These observations can help you predict the last digit even more quickly in certain scenarios. Understanding these patterns not only makes calculations faster but also enhances your number sense and mathematical intuition.

Practical Tips for Mastering Multiplication Last Digits

To get really good at finding the last digit in multiplication problems, here are a few practical tips:

  1. Practice regularly: Like any mathematical skill, finding last digits becomes easier with practice. Try a variety of examples.
  2. Memorize multiplication facts: Knowing your multiplication table up to 10x10 is crucial. This will speed up the process significantly.
  3. Look for patterns: As mentioned earlier, certain patterns can help you quickly determine the last digit. Keep an eye out for these!

By consistently applying these tips and strategies, you can master the art of finding last digits in multiplication problems. Now, let's move on to explore how to find the last digits in expressions involving exponentiation, which adds another layer of complexity but still relies on the same fundamental principles.