Question

A positive integer n, when divided by 9, gives 7 as remainder. What will be the remainder when (3n − 1) is divided by 9?

(a) 1
(b) 2
(c) 3
(d) 4

Answer 1

(b) 2

Let q be the quotient.
It is given that:
Remainder = 7 and Divisor = 9
On applying Euclid's algorithm, we get:
n = 9q + 7 ...(1)
By multiplying both sides by 3, equation (1) becomes:
3n = 27q + 21 ...(2)
By subtracting 1 from both sides, equation (2) becomes:
3n − 1 = 27q + 20
⇒ 3n − 1 = 9 $\times$ 3q + 9 $\times$ 2 + 2
⇒ 3n − 1 = 9 $\times$ (3q + 2) + 2
So, when (3n − 1) is divided by 9, the remainder is 2.