Find the smallest non-negative integer x that satisfies the linear modular congruence 3x \… — JAMB UTME

Find the smallest non-negative integer $x$ that satisfies the linear modular congruence $3x \equiv 8 \pmod{11}$ .

Answer: 10

The smallest non-negative integer

x

is 10.

Evaluating

3x \equiv 8 \pmod{11}

by testing multiples of

11

added to

8

gives

30

, which divided by

3

yields

x = 10

. Since

10 \in [0, 10]

, it is the canonical solution.

Convert the modular congruence into an algebraic equation

3x = 8 + 11k

for an integer

k

By definition of congruence modulo

11

3x - 8

must be a multiple of

11

Find the smallest integer

k \ge 0

such that

8 + 11k

is divisible by 3

When

k = 2

8 + 11(2) = 30

30

is divisible by

3

(

30 / 3 = 10

Divide by 3 to isolate

x

x = 10

3(10) = 30 \equiv 8 \pmod{11}

Linear Modular Congruence

Estimated Time:1m 15s

Question