Q. 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 either of I and II, by itself, is sufficient.

B only II is sufficient, but I is not.

C only I is sufficient, but II is not.

D it is necessary and sufficient to take I and II together.

E even taking I and II together is not sufficient.

## EXPLANATION

The age of Nadeem is a two-digit number. When squared yields a three-digit number whose last digit is Y. Y is a prime number.

Using statement 1:

When a number is squared :

The last digit of the number can be :

1, 2, 3, 4, 5, 6, 7, 8, 9. When squared the last digit can be :

For a number ending with 1: 1

For a number ending with 2: 4

For a number ending with 3: 9

For a number ending with 4: 6

For a number ending with 5: 5

For a number ending with 6: 6

For a number ending with 7: 9

For a number ending with 8: 4

For a number ending with 9: 1

The only possible prime number is 5.

Hence the last digit of X is 5 and Y is 5.

Using statement 2 : Y = X/3.

This alone cannot be sufficient to determine the possibilities for Y and X.

Combining both the statements :

Since Y = 5, then the value of X = 15.

The age is equal to 15.

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