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UPSC CDS I 2025 Mathematics Paper

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Question : 78 of 100

Marks: +1, -0

What is the remainder when

2^{101}

is divided by 101 ?

1
2
5
7

Solution:

Concept:

Use Fermat's Little Theorem: If

p

is prime and

a

is not a multiple of

p

, then

a^{p-1} \equiv 1 \pmod{p}

.

Explanation:

Here,

p = 101

(prime) and

a = 2

(not divisible by 101).

By the theorem,

2^{100} \equiv 1 \pmod{101}

.

We need

2^{101}

. Write

2^{101} = 2 \times 2^{100}

.

Then

2^{101} \equiv 2 \times 1 \equiv 2 \pmod{101}

.

So the remainder when

2^{101}

is divided by 101 is 2.

Answer:

B. 2

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