If x is the smallest positive integer satisfying the modular congruence 2^x \equiv 7 \pmod… — JAMB UTME

If $x$ is the smallest positive integer satisfying the modular congruence $2^x \equiv 7 \pmod{11}$ , what is the value of $(3x^2 - 4x + 5) \pmod{11}$ expressed in standard non-negative remainder form?

Answer: 3

Answer

The smallest positive integer exponent satisfying

2^x \equiv 7 \pmod{11}

x = 7

. Substituting

x = 7

into

3x^2 - 4x + 5

yields

124

, which simplifies to

3 \pmod{11}

Evaluating powers of 2 modulo 11 shows that

2^7 = 128 \equiv 7 \pmod{11}

, giving

x = 7

. Substituting

x = 7

into

3x^2 - 4x + 5

gives

124

, which leaves a remainder of

3

when divided by

11

Step-by-Step Solution

Find the smallest positive integer exponent

x

satisfying

2^x \equiv 7 \pmod{11}

x = 7

Evaluating consecutive powers of 2 modulo 11 shows

2^1 \equiv 2

2^2 \equiv 4

2^3 \equiv 8

2^4 \equiv 5

2^5 \equiv 10

2^6 \equiv 9

, and

2^7 \equiv 7

, making

x = 7

the smallest positive integer power.

Substitute

x = 7

into the expression

3x^2 - 4x + 5

124

Direct substitution gives

3(7)^2 - 4(7) + 5 = 3(49) - 28 + 5 = 147 - 28 + 5 = 124

Reduce 124 modulo 11 to standard non-negative remainder form

Dividing 124 by 11 yields a quotient of 11 with a remainder of 3 (

124 = 11 \times 11 + 3

Key Concept

Modular Exponentiation and Algebraic Evaluation in Modular Arithmetic

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