Which of the following statements can be used to take a decision on this problem ?
What is the product of two natural numbers X and Y ?

Correct option is B

Statement A: Problem cannot be solved.

This is a general statement and does not provide any useful information. It assumes the problem cannot be solved without examining the other options. Hence, it is not sufficient to determine the product of $X$ and $Y$.

Statement B: LCM of $X$ and $Y$ is 71.

• If the LCM of $X$ and $Y$ is 71, it means $X$ and $Y$ must be factors of 71. Since 71 is a prime number, its only factors are 1 and 71.
• This implies $X=1$ and $Y=71$ (or vice versa). The product of $X$ and $Y$ is : $$X\times Y=1\times 71=71.$$
• This statement alone is sufficient to determine the product of $X$ and $Y$.

Statement C: Both $X$ and $Y$ are multiples of 15.

• If both $X$ and $Y$ are multiples of 15, then $X=15a$ and $Y=15b$, where $a$ and $b$ are positive integers.
• The product of $X$ and $Y$ would be:$$X\times Y=(15a)\times (15b)=225\times (a\times b).$$
• However, without additional information about $a$ and $b$, the product cannot be determined uniquely.
• This statement alone is not sufficient.

Statement D: $X$ and $Y$ are distinct.

• Knowing that $X$ and $Y$ are distinct does not provide any numerical values for $X$ or $Y$. It only states that $X\ne Y$, which is not enough to determine their product.
• This statement alone is not sufficient.

Conclusion:

The only statement that is sufficient to solve the problem is Statement B.

Correct Answer: B. LCM of $X$ and $Y$ is 71