Mentor: To see what remainders are, we need to look at division of whole numbers. Suppose we have two
numbers X and Y and we want to see how many times X goes into Y. We would use long division,
and we would get the number of times X can be divided out of Y -- this is called the
quotient -- plus possibly some left over amount if X doesn't divide Y evenly -- this is called the
remainder.

Student: So let me try an example. Suppose I want to find out how many times 3 goes into 14. I know 3
times 4 is 12 and 3 times 5 is 15, so I can get 4 threes out of 14 plus some left?

Mentor: That's right. 4 is the quotient and 2 is the leftover, also called the remainder.

Student: This is related to
mixed number fractions, isn't it?

Mentor: That is one way we can write the answer to this kind of division. The result of dividing 14
by 3 is 4 with a remainder of 2, or 4 and 2/3 as a mixed number, or 4.666666667 as a 10-digit
decimal number.

Try another one.

Student: How about a big one like 67 / 4? Let's see; using my calculator,

How do I get the remainder from this? I don't say 0.75, do I?

Mentor: No, but you have all the information you need. The 16.75 lets you know that 4 goes into 67
sixteen times, so now to find the remainder, just take.

67 - 4 times 16

to get the left over. In this case the remainder is 3.

Student: Couldn't I have figured that out from the 0.75? I mean 0.75 times 4 is 3.

Mentor: Yes! That's a good observation. I like the other method better, though, because it works
exactly even if the calculator rounds. Try 67 / 7. On my calculator, I get:

What is the remainder here?

Student: I see what you mean. My way, I get

0.571428571 times 7 = 3.999999997,

which I could guess means that the answer is 4. Your way I get

67 - 9 times 7 = 4 exactly.

Mentor: Either way gets you the right remainder, you just have to remember to "fix" the calculator
answer to be a whole number. This process of finding quotients and remainders is called the
Euclidean Algorithm after Euclid.

Before we try
Coloring Remainders , we need to talk a little more about remainders. Suppose I turn the question upside down:
What numbers have remainder X when I divide by Y? Try this: What numbers have remainder 2 when
I divide by 3?

Student: Let's see:

Division

Remainder

0/3

none

1/3

1

2/3

2

3/3

none

4/3

1

5/3

2

6/3

none

7/3

1

8/3

2

9/3

none

Neat! I see the pattern. Starting with 2, every third number has remainder 2 again.

Mentor: Very good! Notice that this is true for each possible remainder: 0 (which is what we call "no
remainder" situations), 1 and 2 all repeat. Here is a way to remember this -- mathematicians
call this idea
modular arithmetic. Put your possible remainders in a circle -- this would be all the numbers smaller than your
given divisor -- and walk around the circle counting to find the remainder. Try it for a
different number, say 5:

Student: So to find the remainder when 22 is divided by 5, I start at 0 and count. I get to 2 when
counting 22 moves around the circle.

Mentor: Does this match our other methods?

Student: Seems to: 22/5 is 4.4 and 0.4 times 5 is 2.

Mentor: This idea comes in very handy in advanced mathematics. It is so common that terminology was
invented to ask this question concisely. When we want to find the remainder that occurs when
22 is divided by 5, we ask:
What is 22 mod 5?.

Try one more: What is 8 mod 5?

Student: I like the circle, so I'll do it that way. Counting moves around gets me to 3, so 8 mod 5 is
3.