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Example 9 What are $q$ and $r$ if $a=75$ and $b=24$? If $a=75$ and $b=25$?
The Division Algorithm
Theorem Let $a\in\mathbb{Z}^{+}\cup\{0\}$ and $b\in\mathbb{Z}^{+}$. Then, there exists distinct integers $q,r\in\mathbb{Z}$ where $r\in[0,b)$ and
Recall
i. Least-Integer Principle: A nonempty set of integers that is bounded below contains a smallest element.
ii. Greatest-Integer Principle: a nonempty set of integers that is bounded above contains a largest element.
Let $a=75$ and $b=24$.
Then, $a=bq+r\Rightarrow75=24q+r$.
We can perform long division to see that $75=24(3)+3$ or simply use inspection.
Or check $\frac{75}{24}=\frac{25}{8}=\frac{24+1}{8}=\frac{24}{8}+\frac{1}{8}=3+\frac{1}{8}$.
Then, $75=24\left(3+\frac{1}{8}\right)=24(3)+\frac{24}{8}=24(3)+3$.
Or simply by inspection.
Therefore, $q=3,r=3$.
The ladder half of the question is clear by inspection, $75=25(3)+0$.
Thus, $q=3,r=0$.
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