What is the positive remainder when the expression 7 + 11 \times 24^{137} is divided by 25… — State PSC Exam

What is the positive remainder when the expression $7 + 11 \times 24^{137}$ is divided by $25$ ?

The correct answer is found by substituting the base

24

with

-1

modulo

25

. Since

137

is an odd exponent,

(-1)^{137}

evaluates to

-1

. Following the order of operations, multiply this by

11

to get

-11

, and then add

7

to yield

-4

. Finally, to find the equivalent positive remainder modulo

25

, we add the divisor:

25 - 4 = 21

Express the base

24

in terms of a negative remainder modulo

25

24 \equiv -1 \pmod{25}

Using

-1

simplifies the calculation of extremely large powers without fully expanding them.

Evaluate the exponent part modulo

25

24^{137} \equiv (-1)^{137} \equiv -1 \pmod{25}

An odd power of a negative number retains the negative sign.

Substitute this result back into the original expression and evaluate it.

7 + 11 \times (-1) = 7 - 11 = -4 \pmod{25}

Applying standard order of operations (BODMAS), multiplication must be performed before addition.

Convert the negative intermediate remainder into a positive remainder.

-4 + 25 = 21

A valid remainder in standard division must be non-negative and strictly less than the divisor.

Applying modular arithmetic properties, handling negative remainders appropriately, and maintaining the correct order of algebraic operations.

Soru