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 $a \times b$ is divided by $5$ ?

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Solution

Finding the remainder:
The remainder can be found by using the Division Algorithm theorem.
If $A \div B$ ,where $A$ and $B$ are any integer with $B > 0$ , then there exist unique integers $Q, R$ such that
$A = B Q + R$ and $0 < R < B$ $. . . [1]$
Here we say $Q =$ quotient and
$R =$ remainder
Using the Division Algorithm theorem, When integer $a$ is divided by $5$ the remainder is $2$ and let $q_{1}$ be a quotient.
$a = 5 q_{1} + 2$ $. . . [2]$
Using the Division Algorithm theorem, When integer $b$ is divided by $5$ the remainder is $3$ and let $q_{2}$ be a quotient.
$b = 5 q_{2} + 3$ $. . . [3]$
Multiplying $[1]$ and $[2]$ we get,
$a \times b = (5 q_{1} + 2) (5 q_{2} + 3)$
$a \times b = 25 q_{1} q_{2} + 15 q_{1} + 10 q_{2} + 5 + 1$
$a \times b = 5 (5 q_{1} q_{2} + 3 q_{1} + 2 q_{2} + 1) + 1$
$a \times b = 5 Q + 6$ $. . . [4]$ [Let $Q = (5 q_{1} q_{2} + 3 q_{1} + 2 q_{2} + 1)$ ]
Comparing $[1]$ and $[4]$
Remainder $= 1$
Hence, the remainder is $1$ .

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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×b is divided by 5?

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 $a \times b$ is divided by $5$ ?