Oh, dude, let me break it down for you. So, numbers with 12 factors are like those cool kids at the math party. They're rare, but they exist. The numbers between 1 and 100 that fit the bill are 60 and 84. It's like finding a unicorn in a sea of horses.
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Numbers with 12 factors must be of the form ( p_1^{11} ) or ( p_1^3 * p_2^2 ), where ( p_1 ) and ( p_2 ) are distinct prime numbers. The exponents must be one less than a Prime number. The prime factorization of 12 is 2^2 * 3, so the possible prime factorizations are 2^11 and 2^3 * 3^2. The numbers between 1 and 100 that satisfy these conditions are 2^11 = 2048 and 2^3 * 3^2 = 72.
60, 72, 84, 90 and 96
60, 72, 84, 90 and 96
The maximum number of different factors is 12 and there are 5 numbers in 1-100 which have 12 factors.
108, 126, 132, 140, 150, 156, 160, 198
There are five numbers in that range with 12 factors, but none with exactly eleven.
108, 126, 132, 140, 150, 156, 160, 198, 200 all have 12. Several others have more.