What is the smallest non-negative integer x that satisfies the modular congruence 5x + 7 \… — JAMB UTME

What is the smallest non-negative integer $x$ that satisfies the modular congruence $5x + 7 \equiv 2 \pmod{9}$ ?

The smallest non-negative integer

x

is 8.

Subtracting

7

from both sides yields

5x \equiv -5 \pmod{9}

, which reduces to

5x \equiv 4 \pmod{9}

. Multiplying both sides by the modular multiplicative inverse of

5

modulo

9

(which is

2

, since

5 \times 2 = 10 \equiv 1 \pmod{9}

) gives

x \equiv 4 \times 2 = 8 \pmod{9}

. Thus, the smallest non-negative integer value is

8

Isolate the term containing

x

by subtracting 7 from both sides of the congruence.

5x \equiv 2 - 7 \equiv -5 \pmod{9}

Standard algebraic balance operations apply under modular arithmetic.

Convert the negative remainder

-5

to its canonical non-negative representative in modulo 9.

-5 + 9 = 4 \implies 5x \equiv 4 \pmod{9}

Modular numbers are conventionally expressed in the non-negative range

[0, n-1]

Find the modular multiplicative inverse of 5 modulo 9 and multiply both sides by it.

Since

5 \times 2 = 10 \equiv 1 \pmod{9}

, the inverse is 2. Multiplying gives

x \equiv 4 \times 2 = 8 \pmod{9}

Multiplying by the modular inverse solves for

x

when

\gcd(5, 9) = 1

Linear Modular Congruences and Modular Inverses

Estimated Time:1m 30s

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