Question

On a certain day, the sum of the date and the square root of the month gives the square of the month. Find the date.

• A. 14th February
• B. 16th April
• C. 16th February
• D. 14th April

Hint:

Convert the verbal condition into an equation and check for integer solutions within valid month/day ranges.

Answer 1

The Correct Option is D

Step 1: Given equation

Let the month number be \( m \) and the date be \( d \). We are told: \[ d + \sqrt{m} = m^2 \] Thus: \[ d = m^2 - \sqrt{m} \]

Step 2: Integer condition

Since \( d \) must be an integer, \( \sqrt{m} \) must also be an integer. Therefore, \( m \) must be a perfect square: \[ m = 1, 4, 9 \quad (\text{within 12 months}) \]

Step 3: Testing possible values

• \( m = 1 \): \( d = 1 - 1 = 0 \) → invalid date.
• \( m = 4 \): \( d = 16 - 2 = 14 \) → valid date.
• \( m = 9 \): \( d = 81 - 3 = 78 \) → invalid date.

Step 4: Conclusion

The only valid date is: \[ \boxed{\text{14th April}} \]