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UPSC CDS 1 2021 Math Paper

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Question : 92 of 100

Marks: +1, -0

What is the remainder when

2^{1000000}

is divided by 7?

1
2
4
6

Solution:

Concept:

Use the cyclic property of powers of 2 modulo 7.

Explanation:

We need to find

2^{1000000}\pmod{7}

.

Observe that

2^3 = 8

, and

8 \div 7

leaves remainder

1

.

Thus,

2^3 \equiv 1 \pmod{7}

.

Now write the exponent

1000000

as

3 \times 333333 + 1

.

So

2^{1000000} = 2^{3 \times 333333 + 1} = (2^3)^{333333} \times 2

.

Since

2^3 \equiv 1 \pmod{7}

, we have

(2^3)^{333333} \equiv 1^{333333} \equiv 1 \pmod{7}

.

Therefore,

2^{1000000} \equiv 1 \times 2 \equiv 2 \pmod{7}

.

Hence, the remainder is

2

.

Answer:

B. 2

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