Question

In Euclid's Division Lemma when $a=bq+r$ where $a,b$ are positive integers then what values $r$ can take?

Answer 1

Find the value of the integer

According to Euclid's Division Lemma if we have two integers $a$ and $b$ then there exist unique integer $q$ and $r$ which satisfy the condition $a=bq+r$ where $0\le r<b$

$r$ is the remainder and $b$ is the divisor and remainder is always less than divisor and greater than equal; to $0$

For example $a=59$ and $b=5$

Then according to Euclid's division lemma $a$ can be expressed as

$$\begin{gathered} a=bq+r \\ \Rightarrow 59=11·5+4 \end{gathered}$$

Now take $a=60$ and $b=5$

$$\begin{gathered} a=bq+r \\ \Rightarrow 60=12·5+0 \end{gathered}$$

We can see value of $r$ ranges from $0$ to less than $b$

Hence the value of $r$ is $0\le r<b$