SOLUTION: If 63^n is the greatest power of 63 that divides into 122!, find n.

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Question 1199903: If 63^n is the greatest power of 63 that divides into 122!, find n.

Answer by ikleyn(52848) About Me  (Show Source):
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If 63^n is the greatest power of 63 that divides into 122!, find n.
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63%5En = %289%2A7%29%5En = 3%5E%282n%29%2A7%5En.


The maximum degree  3%5Ek  which divides  122!  is  

      [122%2F3] + [122%2F9] + [122%2F27] + [122%2F81] = 

    =  40    +   13    +   4    +   1     = 58.


  +---------------------------------------------------------------------------+
  | Here  [...]  is the floor-function (the maximum-integer-lesser function). |
  +---------------------------------------------------------------------------+


The maximum degree  7%5Ek  which divides  122!  is  

      [122%2F7] + [122%2F49] = 17  + 2 = 19.


So,  n  must be the greatest integer such that 2n is lesser than or equal to 58
         and (at the same time) be the greatest integer such that n is lesser than or equal to 19.


From these conditions, n = 19.     ANSWER

Solved.