Multiplication Identities - STD 9 - Page 66.1.1 EM

Remainder of Product when Divided by 3

Remainder when Product is Divided by 3

One number leaves remainder 1, the other leaves remainder 2

Problem

Two numbers are such that when divided by 3, one leaves remainder 1 and the other leaves remainder 2. What is the remainder when their product is divided by 3?

First Number (leaves rem 1)
10
10 ÷ 3 = 3 remainder 1
Second Number (leaves rem 2)
14
14 ÷ 3 = 4 remainder 2
Product
140
Remainder when divided by 3
2

Algebraic Method (Why remainder is always 2)

Let the first number = 3m + 1

Let the second number = 3n + 2

Their product = (3m + 1)(3n + 2) = 9mn + 6m + 3n + 2

When divided by 3, 9mn, 6m, and 3n are all divisible by 3.

So the remainder is always 2