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Remainders - A Little Left Over

We teased the exciting world of remainders in the last section. Let's get down to business. When you divide one number by another, sometimes the division is nice and even. Four (4) divided by two (2) gives you the quotient of two (2). But what happens if you divide five (5) by two (2)? Two goes into five two times, but then there is this pesky one (1) left over. That amount is called the remainder. You will get remainders when you are dividing whole numbers and you are not using decimal values in the quotients.

Examples:
5 ÷ 2 = 2 with remainder 1
7 ÷ 2 = 3 with remainder 1
5 ÷ 3 = 1 r 2
- or -

1 r 2 3 ) 5       - 3         2

If you write out the values in a wordy format, you will begin to see how remainders work. You might answer your teacher by saying, "When you divide five (5) by two (2) you get two sets of two and then an extra one (1)."

Example:
5 ÷ 2 = 2 with remainder 1
5 = 2 + 2 + 1
Do you see how you can make two sets of 2 and one set of 1?

Odds and Evens

You know about odd and even numbers. Do you remember the definition? A number is even if it can be divided by two with no remainder. A number is odd if it has a remainder when divided by two.

Problem:
Even or Odd?
0 | Even because 0 ÷ 2 = 0 with no remainder.
3 | Odd because 3 ÷ 2 = 1 with remainder of 1.
888 | Even because 888 ÷ 2 = 444 with no remainder.
67 | Odd because 67 ÷ 2 = 33 with remainder of 1.

Less Than Ten

In the last section we did some division with numbers less than ten. We skipped a few values because they had remainders. Let's fill in some of the blanks now.

3 ÷ 2 = 1 with remainder 1 (2 + 1 = one set of 2 and one set of 1)
5 ÷ 2 = 2 with remainder 1 (2 + 2 + 1)
7 ÷ 2 = 3 with remainder 1 (2 + 2 + 2 + 1)
9 ÷ 2 = 4 with remainder 1 (2 + 2 + 2 + 2 + 1 = four sets of 2 and one set of 1)

4 ÷ 3 = 1 with remainder 1 (3 + 1)
5 ÷ 3 = 1 with remainder 2 (3 + 2)
7 ÷ 3 = 2 with remainder 1 (3 + 3 + 1)
8 ÷ 3 = 2 with remainder 2 (3 + 3 + 2)

5 ÷ 4 = 1 with remainder 1 (4 + 1)
6 ÷ 4 = 1 with remainder 2 (4 + 2)
7 ÷ 4 = 1 with remainder 3 (4 + 3)
9 ÷ 4 = 2 with remainder 1 (4 + 4 +1 = two sets of 4 and one set of 1)

6 ÷ 5 = 1 with remainder 1 (5 + 1)
7 ÷ 5 = 1 with remainder 2 (5 + 2)
8 ÷ 5 = 1 with remainder 3 (5 + 3)
9 ÷ 5 = 1 with remainder 4 (5 + 4)

Do you see any patterns? A remainder can never be larger than the number you are dividing by (divisor). Even if you are dividing a number by fifty-one (51), you can't have a remainder greater than or equal to fifty-one. It doesn't matter what number you use.

Beyond Remainders

We mentioned decimals. For basic math, you will have remainders. As you move forward in math and division, you will learn that remainders are only a starting point. You will eventually learn how to make decimals. Decimals allow you to write number values that are smaller than one. You might notice that we skipped the division problem of two (2) divided by three (3). We will get to values that are less than one in fractions and decimals.

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