How to Do Multiplication and Division in Modular Arithmetic

Learn more about performing modular arithmetic, how it’s related to finding remainders in division, and how it can help you predict the future.

Jason Marshall, PhD
5-minute read
Episode #54

How to Do Multiplication and Division in Modular Arithmetic


In response to the last article on modular arithmetic, math fan Jeff left a comment on the Math Dude’s Facebook page saying that his daughter had been given the problem of figuring out what day of the week it would be a certain number number of days from today. After starting to count off all the days one by one, Jeff introduced her to modular arithmetic…and she was very excited since it made the problem much easier to solve. But what exactly about modular arithmetic made this so much easier? Well, keep on reading because today we’re going to begin to figure this out.

Recap: What is Modular Arithmetic?

Before we answer this question, let’s take a few minutes to finish off the introduction to modular arithmetic that we began in the last article. As you’ll recall, modular arithmetic is a form of arithmetic for integers in which the number line that we count on is wrapped around into a circle whose length is given by a number called the modulus. For example, in arithmetic modulo 12, like what we have when adding numbers on a normal 12-hour clock, a problem like 10 + 5 (mod 12) has the answer 3 (and not 15) since once we count up to 12 we start over at 1 again.

Okay, that’s how addition works in modular arithmetic. But there’s a lot more to arithmetic than just addition, so let’s now take a look at how the other arithmetic operations work in modular arithmetic too.


About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.

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