Question

Nadeem’s age is a two-digit number X, squaring which yields a three-digit number, whose last digit is Y. Consider the statements below: Statement I: Y is a prime number, Statement II: Y is one-third of X. To determine Nadeem’s age uniquely:

• A. only I is sufficient, but II is not.
• B. even taking I and II together is not sufficient.
• C. it is necessary and sufficient to take I and II together.
• D. only II is sufficient, but I is not.
• E. either of I and II, by itself, is sufficient.

Answer 1

The Correct Option is C

To determine Nadeem's age from the given conditions, we need to analyze the statements and test the possibilities. 

Let's denote Nadeem's age as \( X \). Since Nadeem's age is a two-digit number, \( 10 \leq X < 100 \).

When squared, it gives a three-digit number \( X^2 \) where the last digit is \( Y \). Mathematically, \( X^2 \equiv Y \pmod{10} \).

Statement I: \( Y \) is a prime number.

The possible prime numbers that can be the last digit of any integer are 2, 3, 5, and 7.

Statement II: \( Y \) is one-third of \( X \).

This implies \( Y = \frac{X}{3} \) or equivalently \( 3Y = X \).

Now, let's test Statement I:

• If \( X^2 \) ends in 2, possibilities for \( X \) are limited to 4 because \( 4^2 = 16 \). This requires checking if 4 meets other criteria, which it doesn’t as it's not a two-digit number.
• For \( Y = 3 \), \( X \) should end in 7 as \( 7^2 = 49 \), not applicable here because real roots for two expressions should yield the same calculation.
• For \( Y = 5 \), \( X \) should end in 5 because \( 5^2 = 25 \).
• For \( Y = 7 \), missing a correct root in two digits below 100.

Now, include Statement II:

• Combining: if \( Y = 5 \) and \( X = 15 \) (because \( 3 \times 5 = 15 \)), we find.
• Verify: \( X = 15 \rightarrow 15^2 = 225 \). The last digit is indeed 5, meeting both criteria.

Neither statement alone provides a unique age but together, they determine \( X = 15 \) uniquely. Hence, it is necessary and sufficient to take I and II together.

Conclusion: Taking I and II together is necessary and sufficient to determine Nadeem's age uniquely.