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Difficulty: 3/102025 IOQM 2025 (Q31)

Find the remainder when is divided by .

Options:

  • A.

    4

  • B.

    2

  • C.

    3

  • 1

Guide / Hint

Hint 1: Note that can be written as , so .

Hint 2: Raise both sides to the power of 100: .

Hint 3: Simplify , which is 1 since 100 is even.

Solution

Step 1: Analyze the base modulo :

Step 2: Substitute this congruence into the power :

Step 3: Since the exponent is an even integer:

Step 4: Thus:

So the remainder is .

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