How to Calculate 10 mod 3

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  • 0:03 Steps to Solve
  • 1:40 Using a Clock to…
  • 3:57 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will see how to calculate 10 mod 3. After reading about two different processes we can use to work with modulus, we will apply both of these processes to find a general solution to a mod b and to calculate 10 mod 3.

Steps to Solve

Let's learn how to calculate 10 mod 3. You'll be glad to know that this is a fairly simple process. As long as you know the division process and the different parts of a division problem, you can easily calculate a mod b in general. You see, a mod b is simply an expression representing the remainder when we divide a by b.

To clarify this, let's look at the different parts of a division problem. When we divide a number a by a number b, we find how many times b fits into a, call it q, and then we have some number, r leftover, where 0 ≤ r < b. Let's take a look at what we call these different parts.


We see that in the division problem a / b = q remainder r, a is the dividend, b is the divisor, q is the quotient, and r is the remainder. Let's look at how this relates to calculating modulus.

As we said, a mod b is simply an expression representing the remainder when we divide a by b. Therefore, if a / b = q remainder r, then a mod b = r. This leads to the following steps to find a mod b, in general:

  1. First, divide a by b
  2. then, find the remainder: The remainder is a mod b

Pretty easy, huh? Let's do this for 10 mod 3. First, we divide 10 by 3.

10 / 3 = 3 remainder 1

We see that the remainder is 1, which is the second step, and this tells us that 10 mod 3 = 1. Not hard at all!

Using a Clock to Explain Modular Math

Now that we know how to find 10 mod 3, let's dig a little deeper. As we just saw, we can define a mod b as the remainder when a is divided by b. Let's investigate another way to look at this, and that is with a clock. Huh? Yes, a clock.


You probably didn't realize it, but if you keep track of time on a standard 12-hour clock, you've actually been working with modulus 12 since you started telling time! Let's think about this. Suppose it is 11 o'clock, and you want to know what time it will be in 4 hours. If we add 11 + 4, we get 15, but we wouldn't say it's 15 o'clock, we would say it's 3 o'clock because when we get to 12 o'clock, we start over. Also, notice that 15 divided by 12 is 1 remainder 3, and that means that 15 mod 12 = 3. This is no coincidence! Telling time on a standard 12-hour clock is the same as working in modulus 12.

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