The Division Algorithm
Write whole-number division as divisor × quotient + remainder, then check that the remainder is valid.
Lesson focus
Write whole-number division as divisor × quotient + remainder, then check that the remainder is valid.
Read the worked example, then use the practice and quiz to check your understanding.
Start with a familiar division
Think about $137[[div]]12$. The answer is 11 remainder 5 because $12[[cdot]]11=132$ and there are 5 left over.
Write it as an equation: $137=12[[cdot]]11+5$.
Here 137 is the total, 12 is the divisor, 11 is the quotient, and 5 is the remainder.
The division algorithm is a compact way to write the same idea: $a=bq+r$. It means total = divisor × quotient + remainder.
Check that the remainder is valid
When the divisor is positive, the remainder must satisfy $0[[leq]] r < b$. In words: the remainder cannot be negative, and it must be smaller than the divisor.
Example: $137=12[[cdot]]11+5$ is valid because $0[[leq]]5<12$.
$137=12[[cdot]]10+17$ is arithmetically true, but 17 is not a valid remainder when dividing by 12. It is large enough to make another group of 12.
Before you leave
Write a complete mathematical sentence, not only a final number. Check whether your answer matches the condition in the question.
Practice Problems
Attempt each question before opening its answer.
Practice 1
When 245 is divided by 17, write the result in the form $17q+r$.
Show answer
$245=17[[cdot]]14+7$.
So $q=14$ and $r=7$. Check: $0[[leq]]7<17$.
Practice 2
If $n=11q+7$, what remainder does $n+15$ leave when divided by 11?
Show answer
$n+15=11q+22=11(q+2)$.
There is no amount left over, so the remainder is 0.
Interactive Quiz
Check one key idea before you move on.