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

Answer: (A)

The data in both statements I and II together is necessary to answer the question.

Answer 2

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.