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UPSC CDS I 2025 Mathematics Paper

Question : 48 of 100

Marks: +1, -0

24^{3} - 16^{3} - 8^{3}

Solution:

Concept:

Use factorization by taking common powers and then prime factorization to count the total number of factors.

Explanation:

Factor out

8^{3}

from the expression:

24^{3} - 16^{3} - 8^{3} = 8^{3}(3^{3} - 2^{3} - 1^{3})

= 8^{3} (27 - 8 - 1) = 8^{3} \times 18

Now compute

8^{3} = 512

and

512 \times 18 = 9216

Find the prime factorization of 9216:

9216 = 2^{10} \times 3^{2}

Number of factors is given by

(10+1)(2+1) = 11 \times 3 = 33

Answer:

Option A: 33

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