If x is the least positive integer satisfying the modular congruence 4x + 9 \equiv 3 \pmod… — JAMB UTME

If $x$ is the least positive integer satisfying the modular congruence $4x + 9 \equiv 3 \pmod{11}$ , what is the value of $(x^2 - 5x + 2) \pmod{11}$ expressed in standard non-negative remainder form?

A
2
B
4
C
7
9Answer

Answer

The value of

(x^2 - 5x + 2) \pmod{11}

in standard non-negative remainder form is 9.

Solving

4x + 9 \equiv 3 \pmod{11}

yields

4x \equiv -6 \equiv 5 \pmod{11}

. Multiplying by 3 (the inverse of 4 mod 11) gives

x \equiv 15 \equiv 4 \pmod{11}

. Evaluating

(x^2 - 5x + 2)

x = 4

gives

16 - 20 + 2 = -2

. Converting

-2

into the standard non-negative remainder range

[0, 10]

gives

-2 + 11 = 9

Step-by-Step Solution

Isolate the variable term in the linear modular congruence

4x \equiv 3 - 9 \equiv -6 \pmod{11}

Subtract 9 from both sides of the congruence.

Convert the negative right-hand side to a non-negative residue modulo 11

4x \equiv -6 + 11 \equiv 5 \pmod{11}

Add the modulus 11 to obtain the canonical non-negative equivalent.

Solve for

x

by multiplying by the multiplicative inverse of 4 modulo 11

Since

4 \times 3 = 12 \equiv 1 \pmod{11}

, multiply both sides by 3:

x \equiv 5 \times 3 = 15 \equiv 4 \pmod{11}

. Thus, the least positive integer is

x = 4

The modular inverse of 4 modulo 11 is 3.

Substitute

x = 4

into the expression

(x^2 - 5x + 2)

4^2 - 5(4) + 2 = 16 - 20 + 2 = -2

Evaluate the quadratic expression using the calculated value of

x

Express

-2

in canonical non-negative remainder form modulo 11

-2 + 11 = 9 \pmod{11}

Add the modulus 11 to convert the negative result to a non-negative remainder within

[0, 10]

Key Concept

Solving linear modular congruences and converting negative remainders to canonical form

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