Question

The question given below consists of a question and two statements numbered I and II. You have to decide whether the data provided in the statements is sufficient to answer the question. Read both statements and answer accordingly.
What is the product of x and y?
Statement I. Both x and y are consecutive even multiples of 32.
Statement II. The LCM of x and y is 3584.

• A. The data in both statements I and II together is necessary to answer the question.
• B. The data either in Statement I alone or in Statement II alone is sufficient to answer the question.
• C. The data in both statements I and II together is not sufficient to answer the question.
• D. The data in Statement II alone is sufficient to answer the question, while the data in Statement I alone is not sufficient to answer the question.
• E. The data in Statement I alone is sufficient to answer the question, while the data in Statement II alone is not sufficient to answer the question.

Answer 1

The Correct Option is A

Statement I: Both \(x\) and \(y\) are consecutive even multiples of \(32\).
So, if \(x = 32(2k) = 64k\), then \(y = 32(2k + 2) = 64(k + 1)\)
Product of x and \(y = 64k(64k + 64) = 4096k(k + 1)\)
We do not know the value of \(‘k’\), so we cannot find the product.
Thus, Statement I alone is not sufficient to answer the question.

Statement II: The LCM of x and \(y\) is \(3584\).
Let x and \(y\) be \(‘ha’\) and \(‘hb’\), respectively, where \(‘h’\) is the \(HCF\) of \(x\) and \(y\), and \(‘a’\) and \(‘b’\) are co-primes.
Thus, \(hab = 3584\)
Product of \(x\) and \(y\) = \(ha(hb) = h^2(ab) = 3584\;h\)
We do not know the value of \(‘h’\), so we cannot find the product of \(x\) and \(y\).
Using statements I and II together,
\(LCM \) of \(x\) and \(y\) = \(64k(k + 1) = 3584\)
or, \(k(k + 1) = 56\)
or, \(k = 7\)
Now, the product of \(x\) and \(y\), i.e., \(4096k(k + 1)\) can be calculated.
Thus, both the statements are necessary to answer the question.

Hence, option \(A\) is the correct answer.