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UPSC CDS 1 2022 Math Paper

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Question : 36 of 100

Marks: +1, -0

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

is divisible by

7
9
11
17

Solution: 👈: Video Solution

Given:

The number is

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

Concept Used:

If

n = a \times b \times c \ldots

Then

n

must be divisible by a, b, c, ....

Calculation:

We have

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

Taking

4^{61}

common from this number, we get

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

\Rightarrow 4^{61}(1+4+16+64)

\Rightarrow 4^{61}(85)

\Rightarrow 4^{61} \times 5 \times 17

So, We can write

4^{61}+4^{62}+4^{63}+4^{64}=4^{61} \times 5 \times 17

So, the number can be divisible by 5,17 and

4^{n} \{ n \le 61 \}

From option we can say that

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

is divisible by 17

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