COMEDK · Maths · 20. Sets and Relations

Let \(X\) and \(Y\) be the set of all positive divisors of 400 and 1000 respectively (including 1 and the number). Then \(n(X \cap Y)\) is equal to

A 8
B 12
C 6
D 10

Verified Solution

Answer & Solution

Correct Answer

(B) 12

Step-by-step Solution

Detailed explanation

The prime factorization of 400 is \(400 = 4 \times 100 = 2^2 \times 2^2 \times 5^2 = 2^4 \times 5^2\). The prime factorization of 1000 is \(1000 = 10^3 = 2^3 \times 5^3\). The set \(X\) contains all divisors of \(2^4 \times 5^2\), which are of the form \(2^a \times 5^b\) where…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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