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General Studies Math: Help & Review8 chapters | 85 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Modular arithmetic is something we use everyday when we look at a clock. In this lesson, we'll look at some rules and properties of addition, subtraction, and multiplication in modular arithmetic and how to apply them.

Before we get to the rules and properties of modular arithmetic, let's first review the meaning of the term. **Modular arithmetic**, or clock arithmetic, is something we use on a daily basis. In a regular clock, such as the one pictured here, civilians tell time according to two equally spaced intervals composed of 12 hours, or integers. In modular arithmetic, whole numbers, or **integers**, repeat around a designated number known as the modulus.

In traditional clocks and watches, the **modulus**, or the number at which we begin again, is 12. For example, if it's 10AM, and you're meeting friends four hours later, you'll most likely say you're meeting them at 2PM, not fourteen o'clock. Now, if we were working in modulus 7, then we would start over at 7. So, 6 + 5 in modulus 7 would be 4, not 11.

Another way to look at this is with remainders. When we're working in modulus *n*, then any number in modulus *n* is equal to the remainder when that number is divided by *n*. Consider our modulus 7 example: 6 + 5 = 11. When we divide 11 by modulus 7, we get 1 with a remainder of 4; therefore, 6 + 5 = 4.

Notice that when dealing with modular arithmetic, there are many numbers that are equivalent for a given modulus. Consider the number 5 in modulus 12:

- 5 + 12 = 17
- 17 + 12 = 29
- 29 + 12 = 41

Here, 5 is equivalent to 17, 29, 41, and so on. We can express these equivalencies by saying the numbers are congruent in modulus 12, and write them using a 3-bar equal sign as shown here:

You can perform many of the same operations with modular math that you can with regular math. Here are some rules for addition in modular arithmetic:

Based on these rules, we can either add the numbers together, and then find the sum in modulus *n*. Alternatively, we can first find each of the numbers in modulus *n*, and then add them together. Let's try working in modulus 8.

What is (4 + 7 + 6 + 8) mod 8?

- Add the numbers in the parentheses: 4 + 7 + 6 + 8 = 25
- To find 25mod8, perform the division: 25 / 8 = 3 with a remainder of 1
- Therefore, (4 + 7 + 6 + 8)mod8 is congruent to 25mod8, which is congruent to 1mod8

The following rules can be used to perform subtraction in modular arithmetic:

Again, we have two choices. We can subtract the numbers first, and find modulus *n*. Or we can find each of the numbers in modulus *n* first, and then perform the subtraction. In this example, let's work in modulus 5.

What is (104 - 53)mod5?

- Subtract the numbers in the parentheses: 104 - 53 = 51. Here, (104 - 53)mod5 is congruent to 51mod5.
- Perform the division: 51 / 5 = 10 with a remainder of 1. So, (104 - 53)mod5 is congruent to 1mod5.
- Alternatively, we could first find 104mod5 and 53mod5, or 4mod5 and 3mod5 respectively, and then subtract: 4mod5 - 3mod5 is congruent to 1mod5.

When dealing with subtraction, you may end up with a negative number. For instance, suppose it's 4:00AM or PM, and we want to know what time it was 11 hours ago, or (4 - 11)mod12 = -7mod12. To find how the result is congruent to mod12, we add multiples of 12 until we get to a number that falls between 0 and 11. -7 + 12 = 5

Here, -7mod12 is congruent to 5mod12, which means it was 5:00AM or PM 11 hours ago. In general, when you get a negative number and you're working in modulus *n*, add multiples of *n* to the negative number until you get a number between 0 and *n* - 1.

We can use the following rules to perform multiplication in modular arithmetic:

Here, we can perform the multiplication, and then find that number in modulus *n*. Or, we can find each number in modulus *n* first, and then multiply them. Let's work in modulus 18.

What is (14 * 20)mod18?

- Multiply the numbers in the parentheses: 14*20 = 280mod18
- Perform the division: 280 /18 = 15 with a remainder of 10. So, (14*20)mod18 is congruent to 10mod18
- We could also first find 14mod18 and 20mod18, or 14mod18 and 2mod18 respectively, and then multiply the results. Here, (14mod18) * (2mod18) is congruent to 28mod18. When we divide 28 by 18, we get 1 with a remainder of 10; so, (14 * 20)mod18 is congruent to 10mod18.

**Modular arithmetic** is also called clock arithmetic because the rules are similar to the traditional way we tell time. In modular arithmetic, we have a **modulus**, which is the **integer**, or whole number, at which we start over.

When adding, subtracting, or multiplying in modulus *n*, we can do one of two things:

- We can perform the operation first, and then find that number in modulus
*n*by dividing it by*n*and identifying the remainder - We can also find each number in modulus
*n*individually, and then perform the operation on those numbers

Either way, we'll get the same answer. Don't forget this other important rule: when the result is a negative number, add multiples of the modulus until you end up with a number between 0 and the modulus minus 1. That will be your answer.

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General Studies Math: Help & Review8 chapters | 85 lessons

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