Which Numbers Are Less Than 100 and Have Exactly Three Factors?
Numbers less than 100 that have exactly three factors are four, nine, 25 and 49. All of these numbers are squares of prime numbers, which means that their only factors are one, themselves and their square roots. Two squared is four, three squared is nine, five squared is 25 and seven squared is 49.
The factors of four are one, two and four. The factors of nine are one, three and nine. The factors of 25 are one, five and 25. The factors of 49 are one, seven and 49. The middle number of each factor list is used twice because that factor is squared, but since the factor is the same number, it is not counted twice.
Squares of prime numbers are the only numbers that have exactly three factors. The other numbers under 100 with odd numbers of factors are one, 16, 36 and 81. The numbers 16 and 81 have five factors, while 36 has nine factors. All three of these numbers are exponentially larger than a smaller positive integer.
The number one has exactly one factor, which is itself. One is a special number because it is not prime and has an odd number of factors. Prime numbers have exactly two factors: the number itself and one. The next prime number more than nine, 11, has a square of 121.