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UPSC CDS 1 2022 Math Paper

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Question : 4 of 100

Marks: +1, -0

4^{61} + 4^{62} + 4^{63} + 4^{64}

is divisible by

7
9
11
17

Solution:

Concept:

Factor out the smallest power of

4

to simplify the sum and find its factors.

Explanation:

The given expression is

4^{61} + 4^{62} + 4^{63} + 4^{64}

.

Take the common factor

4^{61}

out:

4^{61}(1 + 4 + 4^{2} + 4^{3})

.

Calculate the bracket:

1 + 4 + 16 + 64 = 85

.

So the expression becomes

4^{61} \times 85

.

Factor

85

as

5 \times 17

.

Thus the number is

4^{61} \times 5 \times 17

.

Therefore, it is divisible by

5

and

17

.

Among the given options,

17

is the correct divisor.

Answer:

17

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