When integer $a$ is divided by $5$ the remainder is $2$ .
When integer $b$ is divided by $5$ the remainder is $3$ .
What is the remainder when $a x b$ is divided by $5$ ?

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Solution

Finding the remainder where $a x b$ is divided by $5$ :
Given,
$\frac{a}{5} =$ Remainder $2$ .
$\frac{b}{5} =$ Remainder $3$ .
We know that,
Dividend $=$ Divisor $x$ Quotient $+$ Remainder
Let we choose $x$ as quotient for dividend $a$ and $y$ as quotient for dividend $b$ .
Divisor $= 5$
So,
$a = 5 x + 2 b = 5 y + 3$ [ by given
Now,
$a x b = (5 x + 2) (5 y + 3) = 25 x y + 10 y + 15 x + 6 = 25 x y + 10 y + 15 x + (5 \times 1) + 1$
$\frac{a x b}{5} =$ remainder $1$ . because the term $25 x y + 10 y + 15 x + (5 \times 1)$ are multiples of $5$ .
Hence, $1$ is the remainder when $a x b$ is divided by $5$ .

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