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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 (noninteractive) 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 $2218 = 4$ 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 $2$ 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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