12 Is a Remarkable Number
 
                                                                           David W. Hansen
                                                                                      © 2008
    
       Its prime factors consist of two of the first prime and one of the second prime; that is,
    
                                          12  =  2(1st prime) x 1(2nd prime)  =  2(2) x 1(3) =   22 x 31
 
       12  =  2 x 6, which is the product of the first prime and the first perfect number.

 

       12  =  1! x 2! x 3! 

 

              =  1 x 2  +  2 x 2  +  3 x 2

 

              =             31  +  32        

 

              =          10  +  21  +  32

 
              =          13  x  22  x  31  
      
        The number of its divisors is 6, (1, 2, 3, 4, 6, 12), which is the first perfect1 number.
 
       The sum of its divisors is 28, (1 + 2 + 3 + 4 + 6 + 12 = 28), which is the second perfect number.
 
       The sum of its proper1 divisors (1 + 2 + 3 + 4 + 6 = 16) is a perfect square (42) and a perfect fourth
        power (24).
 
       The product of its proper divisors (1 x 2 x 3 x 4 x 6  =   144  =  122)  is the square of 12 itself.
 
       The product of its divisors     (1 x 2 x 3 x 4 x 6 x 12  = 1728  =  123)  is the  cube  of 12 itself.
 

        It is the first (smallest) natural number with six divisors.

 
        It is the sum plus the product of the proper factors of the first perfect number 6.
 
                                                              (1 + 2 + 3) +  (1 x 2 x 3)   =   12
 
        It is both a nice2 number and a very nice2 number; in fact, it is the smallest nice number and the
        smallest very nice number.
 
        It is twice the first perfect number, [6 = 1 + 2 + 3, 12 = 2(6)].
 

       Here are some interesting properties of the factors of 12:

 

       a)                                3  x  4      =  12

 

                                         3!  x  4!     =  122,            since       3!  x  4!      =     6 x 24     =    144   =  122            

 

                                     (3!)2  x  (4!)2  =  124,           since  (3!)2  x (4!)2   =   (3! x 4!)2  =  (122)2  =  124    

 

                                     (3!)3  x  (4!)3  =  126,           since  (3!)3  x (4!)3   =   (3! x 4!)3  =  (122)3  =  126    

 

        and                       (3!)n  x  (4!)n  =  122n,         since  (3!)n  x (4!)n   =   (3! x 4!)n  =  (122)n  =  122n    

 

        for n any positive integer.

 

        b)       2  x  6          =   12

 

                 2!  x  6!        =   122  x  10,       since     2!  x  6!     =   2 x 720   =     1440        =   122 x 10

 

             (2!)2  x  (6!)2   =  124  x  102,      since  (2!)2  x (6!)2  =  (2! x 6!)2  =  (122 x 10)2  =   124 x 102

 

             (2!)3  x  (6!)3   =  126  x  103,      since  (2!)3  x (6!)3  =  (2! x 6!)3  =  (122 x 10)3  =   126 x 103

 

             (2!)n  x  (6!)n   =  122n  x  10n,     since  (2!)n  x (6!)n  =  (2! x 6!)n  =  (122 x 10)n  =  122n x 10n

 

        for any positive integer n.

       ______________________________________________________________________________
 
        1 A perfect number is one which is equal to the sum of its proper divisors. A proper divisor of a number is any 
           divisor of the number except itself.   Thus the first perfect number is  6 =1 + 2 + 3, and the second perfect
           number is 28 = 1 + 2 + 4 + 7 + 14.
 
        2 A nice number is a number which can be written as a sum of consecutive integers, a sum of consecutive
           even integers, and a sum of consecutive odd integers. A very nice number is a number which can be
           written as a sum of consecutive integers, a sum of consecutive even integers, and a sum of consecutive
           odd integers in one and only one way.
 
 

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