Understanding Factors and Perfect Squares
When discussing the relationship between the number of factors of a number \( N \) and its status as a perfect square, it's crucial to delve into the properties of factors.

Factors of a Number
- A factor is a number that divides another number without leaving a remainder.
- The total number of factors of a number can be determined using its prime factorization.

Prime Factorization and Count of Factors
- If \( N \) can be expressed as \( N = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \ldots \times p_k^{e_k} \), where \( p_i \) are prime factors and \( e_i \) are their respective powers, the formula for calculating the total number of factors \( T(N) \) is:
\[
T(N) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots (e_k + 1)
\]

Odd Number of Factors
- For \( T(N) \) to be odd, each term \( (e_i + 1) \) in the product must also be odd.
- This condition is satisfied only when all \( e_i \) are even numbers.

Conclusion: Perfect Square
- Since each exponent \( e_i \) being even indicates that \( N \) can be expressed as \( N = p_1^{2k_1} \times p_2^{2k_2} \times \ldots \times p_k^{2k_k} \), it follows that \( N \) is indeed a perfect square.
- Therefore, if a number \( N \) has an odd number of factors, it must be a perfect square.
This relationship highlights the fascinating interplay between number theory concepts and the properties of integers. Understanding this can aid in solving various mathematical problems, particularly in competitive exams like UPSC.