Question

When integer $a$ is divided by $5,$ the remainder is $2$. When integer $b$ is divide by $5,$ the remainder is $3$. What is the remainder when $a\times b$ is divide by $5$.

Answer 1

Step 1: Find the values of $a$ and $b$.

Given that, when integer $a$ is divided by $5,$ the remainder is $2$.

By division algorithm,

$\mathrm{Dividend}=\left(\mathrm{quotient}\times \mathrm{divisor}\right)+r\mathrm{emainder}$

Let $x$ be the quotient so, $a=5x+2$

Also given that, When integer $b$ is divide by $5,$ the remainder is $3$.

Let $y$ be the quotient so, $b=5y+3$.

Step 2: Find the remainder.

$$\begin{gathered} a\times b=\left(5x+2\right)\left(5y+3\right) \\ \Rightarrow a\times b=25xy+15x+10y+6 \\ \Rightarrow a\times b=25xy+15x+10y+5+1 \\ \therefore a\times b=5(5xy+3x+2y+1)+1 \end{gathered}$$

When $a\times b$ is divided by $5$.

Let $z$ be the quotient and $r$ be the remainder.

So, $a\times b=5z+r$.

Comparing both equations, $a\times b=5(5xy+3x+2y+1)+1$ and $a\times b=5z+r$, we get

$r=1$.

Therefore, the remainder is $1$.

Hence, when $a\times b$ is divided by $5$ the remainder is $1$.