What is the remainder when the expression 4 + 9 \times 11^{85} is divided by 12? — State PSC Exam

What is the remainder when the expression $4 + 9 \times 11^{85}$ is divided by $12$ ?

The correct remainder is 7.

The correct answer is 7 because applying modular arithmetic rules simplifies

11^{85}

-1

modulo 12. Substituting this back into the expression yields

4 + 9 \times (-1) = -5

. To find the true positive remainder, the modulus 12 is added to

-5

, resulting in exactly 7.

Find the remainder of the base 11 when divided by the modulus 12.

11 \equiv -1 \pmod{12}

Using negative remainders simplifies calculations involving large exponents.

Evaluate the exponentiation term modulo 12.

11^{85} \equiv (-1)^{85} = -1 \pmod{12}

A negative number raised to an odd power remains negative.

Substitute the evaluated power back into the expression following the standard order of operations (BODMAS).

4 + 9 \times (-1) = 4 - 9 = -5 \pmod{12}

Multiplication must be performed before addition.

Convert the negative remainder into a valid positive remainder.

-5 + 12 = 7

The final remainder must be a positive integer between 0 and 11, achieved by adding the modulus.

Modular Arithmetic, Negative Remainders, and Order of Operations

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