What is the canonical non-negative remainder when 3^{45} - 11 is divided by 7? — JAMB UTME

What is the canonical non-negative remainder when $3^{45} - 11$ is divided by $7$ ?

Answer: 2

The correct answer is 2.

Using modular exponentiation,

3^3 = 27 \equiv -1 \pmod{7}

, so

3^{45} = (3^3)^{15} \equiv (-1)^{15} = -1 \pmod{7}

. Subtracting 11 yields

-1 - 11 = -12 \pmod{7}

. Adding 14 (a multiple of 7) gives

-12 + 14 = 2

, which is the canonical non-negative remainder.

Evaluate

3^{45} \pmod{7}

using power rules

3^{45} \equiv -1 \pmod{7}

Since

3^3 = 27 \equiv -1 \pmod{7}

, raising both sides to the power of 15 gives

(3^3)^{15} \equiv (-1)^{15} = -1 \pmod{7}

Subtract 11 from the modular result

-12 \pmod{7}

Replacing

3^{45}

with

-1

gives

-1 - 11 = -12

Reduce -12 to its canonical non-negative remainder in

[0, 6]

Adding the smallest multiple of 7 that makes the result non-negative gives

-12 + 14 = 2

Modular Exponentiation and Negative Remainder Reduction

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