Multiplication Identities - STD 9 - Page 66.1.2 EM

Remainder of Product when Divided by 4

Remainder when Product is Divided by 4

One number leaves remainder 1, the other leaves remainder 2

Problem

Two numbers are such that when divided by 4, one leaves remainder 1 and the other leaves remainder 2. What is the remainder when their product is divided by 4?

First Number (leaves rem 1)
9
9 ÷ 4 = 2 remainder 1
Second Number (leaves rem 2)
10
10 ÷ 4 = 2 remainder 2
Product
90
Remainder when divided by 4
2

Algebraic Method (Why remainder is always 2)

Let the first number = 4m + 1

Let the second number = 4n + 2

Their product = (4m + 1)(4n + 2) = 16mn + 8m + 4n + 2

When divided by 4, 16mn, 8m, and 4n are all divisible by 4.

So the remainder is always 2