Two Statements S1 and S2 are given below with regard to two numbers followed by a Question:

S1: Their product is 21.

S2: Their sum is 10.

Question: What are the two numbers?

Which one of the following is correct in respect of the above Statements and the Question?

The correct option is: S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question.

Let the two unknown numbers be $x$ and $y$.

Step 1: Analyze Statement 1 (S1) alone

Statement S1 states that the product of the two numbers is 21:

$x\times y=21$

There are multiple pairs of numbers whose product is 21 (for example: 1 and 21, 3 and 7, or non-integer pairs like 2 and 10.5). Thus, S1 alone is not sufficient to uniquely determine the two numbers.

Step 2: Analyze Statement 2 (S2) alone

Statement S2 states that the sum of the two numbers is 10:

$x+y=10$

There are infinitely many pairs of numbers whose sum is 10 (for example: 1 and 9, 2 and 8, 3 and 7, etc.). Thus, S2 alone is not sufficient to uniquely determine the two numbers.

Step 3: Analyze Statements 1 and 2 together

Combining both S1 and S2 gives us a system of two equations:

1) $xy=21$

2) $x+y=10$

From equation (2), we can express $y$ in terms of $x$:

$y=10-x$

Substitute $y$ into equation (1):

$x(10-x)=21$

$10x-x^2=21$

$x^2-10x+21=0$

Factoring the quadratic equation:

$(x-3)(x-7)=0$

This yields $x=3$ or $x=7$. Correspondingly, the two numbers are uniquely identified as 3 and 7.

Conclusion:

Both statements together are required and sufficient to answer the Question, while neither statement alone provides enough information.