
Shortcut Trick
Exactly 3 factors occur only when a number is a square of a prime number (p2).
Factors of p2 are 1, p, and p2.
Prime numbers (p) such that p2 < 100 are 2, 3, 5, and 7.
Squares: 22=4, 32=9, 52=25, 72=49.
The numbers are {4, 9, 25, 49}.
∴ The correct answer is 4.

Alternate Method
Given: Numbers less than 100 with exactly 3 factors.
Formula Used: If N = p1a × p2b, total factors = (a + 1)(b + 1).
Numbers with Exactly 3 Factors (Squares of Primes) 2²=4 3²=9 5²=25 7²=49 Total such numbers = 4
⇒ Total factors required = 3
⇒ (a + 1) = 3 ⇒ a = 2
⇒ The number must be of the form p2, where p is a prime number.
⇒ List squares of prime numbers:
⇒ 22 = 4 (Factors: 1, 2, 4)
⇒ 32 = 9 (Factors: 1, 3, 9)
⇒ 52 = 25 (Factors: 1, 5, 25)
⇒ 72 = 49 (Factors: 1, 7, 49)
⇒ 112 = 121 (This is greater than 100)
⇒ Total count = 4
∴ The correct answer is 4.

Additional Information
Number of Factors Formula
If N = pa × qb × rc, total factors = (a+1)(b+1)(c+1), where p, q, r are prime factors.
Perfect Squares
A perfect square always has an odd number of factors (e.g., 16 has factors 1, 2, 4, 8, 16 which counts to 5).
Prime Numbers
A prime number has exactly 2 factors: 1 and itself.
Composite Numbers
Composite numbers have more than 2 factors. 1 is neither prime nor composite.