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# Remainders and Quotients

This module is about integers (whole numbers), dividing, quotients and remainders. It is useful when programming and using type `int`

(short for integer). For example, few people would say that 17 days is $2\frac37$ weeks. Instead we stay with integers and divide 17 by 7. We find that 17 days is 2 weeks and 3 days.

The contents of these pages can be downloaded as a (non-interactive) PDF file from here.

Here are some examples.

##### Example 1

I'm packing eggs into boxes which hold $6$ eggs each. I have $22$ eggs.

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How many boxes can I fill and how many eggs do I have left over?

**Answer.** Using $3$ boxes I can pack $3\times 6=18 $ eggs.

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I have $22-18 = 4$ eggs left over. So the answer is $3$ boxes with $4$ eggs left over.

In this context I would not say that I'd packed $3\frac46$ or $3\frac23$ boxes.

Here are some similar questions.

##### Example 2

How many years and months is $18$ months?

**Answer:** It is $1$ year and $6$ months. We could say it's $1.5$ years, but we want a whole number of years together with the number of months left over.

##### Example 3

How many minutes and seconds is $134$ seconds?

**Answer:** Each minute is $60$ seconds, and $134$ is $2\times 60+14$, so it's $2$ minutes and $14$ seconds.

We would not say that it's $1$ minute and $74$ seconds, as $74$ seconds is more than a minute.

##### Example 4

How many weeks and days is $25$ days?

**Answer:** It's $3$ weeks and $4$ days because $25 = 7\times3 + 4$.

In each of these questions we have taken a number, divided it by another number to get the **quotient** and then worked out how much was left over, the **remainder**.

In Example 1 we divided $22$ by $6$. We found that $22$ divided by $6$ is $3$ with remainder $4$. The quotient is $3$.

We could also say "$22$ divided by $6$ is $3$ with $4$ left over" or `"$6$ goes into $22$ $3$ times with remainder $4$". If we were happy to work with fractions we could say "$22$ divided by $6$ is $3\frac46$", but that doesn't tell us directly how many boxes were filled and how many eggs were left.

In general, for integers (whole numbers) $a$ and $b$ with $b>0$, we say that $a$ divided by $b$ is $q$ with remainder $r$. This means that

$$a=b\times q+r,$$

$q$ and $r$ are integers with $0\le r$

"$b$ goes into $a$, $q$ times with remainder $r$"

"$a$ divided by $b$ is $q$ with remainder $r$".

There is only one correct quotient and remainder. The remainder is always $0$ or positive; it is also always less than $b$. In Example 1. we divided $22$ by $6$:

and so $22$ divided by $6$ is $3$ with remainder $4$. The quotient is $3$ and the remainder is $4$.

In Example 2, $18$ was divided by $12$; the quotient is $1$ and the remainder $6$:

To determine whether a number is even or odd, look at the remainder after it has been divided by $2$: an even number is one which has remainder $0$ and an odd number is one with remainder $1$. For example, as $17 = 2\times 8 + 1$ (so the remainder is $1$), $17$ is odd.

Now try these problems:

**How many times does $6$ go into $50$?**

**How many times does $2$ go into $30$? Is $30$ even or odd?**

**Find the quotient and remainder when $23$ is divided by $5$.**

**This one is to make you think. How many times does $23$ go into $5$ and what is the remainder?.**

**How many times does $5$ go into $35$ and what is the remainder?.**

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