If the number 11^{2025} is decreased by 16^{2025}, and the resulting difference is divided… — State PSC Exam

If the number $11^{2025}$ is decreased by $16^{2025}$ , and the resulting difference is divided by $9$ , what is the remainder?

To find the remainder of

11^{2025} - 16^{2025}

divided by

9

, we first simplify the bases using modular arithmetic:

11 \equiv 2 \pmod 9

and

16 \equiv -2 \pmod 9

. Next, we evaluate the powers. We know

2^3 = 8 \equiv -1 \pmod 9

. By rewriting the exponent,

2^{2025} = (2^3)^{675} \equiv (-1)^{675} = -1 \pmod 9

. For the second term,

(-2)^{2025} = - (2^{2025}) \equiv -(-1) = 1 \pmod 9

. Subtracting the two terms yields

-1 - 1 = -2

. Because a remainder must be positive, we add the divisor to the negative result:

-2 + 9 = 7

Reduce the base numbers modulo 9.

11 \equiv 2 \pmod 9

and

16 \equiv 7 \equiv -2 \pmod 9

Reducing the bases simplifies the expression to

2^{2025} - (-2)^{2025} \pmod 9

Determine the cyclicity pattern for powers of 2 modulo 9.

2^1=2

2^2=4

2^3=8 \equiv -1 \pmod 9

Finding a power that leaves a remainder of

1

-1

allows us to easily compute extremely large exponents.

Evaluate the first term

2^{2025} \pmod 9

2^{2025} = (2^3)^{675} \equiv (-1)^{675} = -1 \pmod 9

Breaking the exponent

2025

into

3 \times 675

utilizes the known value of

2^3 \pmod 9

Evaluate the second term

(-2)^{2025} \pmod 9

(-2)^{2025} = - (2^{2025}) \equiv -(-1) = 1 \pmod 9

An odd power preserves the negative sign, allowing substitution of the result from the previous step.

Subtract the remainders and map to a positive equivalent.

(-1) - 1 = -2

. To make it positive, add

9

-2 + 9 = 7

Remainders must be strictly positive and less than the divisor. Adding the divisor to a negative modular result finds its positive equivalent.

Modular Arithmetic Cyclicity and Negative Remainders

Question