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UPSC CDS II 2024 Math Paper

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Question : 91 of 100

Marks: +1, -0

What is the HCF of 2^{36 }- 1 and 2^{45 }- 1?

1023
512
511
255

Solution:

Concept:

The HCF of two numbers of the form

2^{m} - 1

and

2^{n} - 1

is

2^{\text{HCF}(m,n)} - 1

.

Explanation:

We need the HCF of

2^{36} - 1

and

2^{45} - 1

.

First, find the HCF of the exponents:

36

and

45

.

Prime factors:

36 = 2^{2} \times 3^{2}

,

45 = 3^{2} \times 5

.

Common factors:

3^{2} = 9

. So

\text{HCF}(36,45) = 9

.

Now apply the formula:

2^{\text{HCF}(36,45)} - 1 = 2^{9} - 1

.

Compute:

2^{9} = 512

, so

512 - 1 = 511

.

Answer:

511

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