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Next year vs the current year (Posted on 2015-02-25) Difficulty: 2 of 5
2016k divides 2016!2016.

For how many positive values of the integer k is the above true?

What would be your answer if the number 2016 was replaced by 2015 in all 2016's appearances?

No Solution Yet Submitted by Ady TZIDON    
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Solution with some computer assistance | Comment 2 of 5 |
2016 is 2^5 * 3^2 * 7^1.

2016! has 2 as a factor 2010 times; 3 as a factor 1004 times; and 7, 334 times.

(the above was determined by

  n = 2016
  For i = 1 To n
    ii = i
    While ii Mod 2 = 0
     fct2 = fct2 + 1
     ii = ii / 2
    Wend
    While ii Mod 3 = 0
     fct3 = fct3 + 1
     ii = ii / 3
    Wend
    While ii Mod 7 = 0
     fct7 = fct7 + 1
     ii = ii / 7
    Wend
    
  Next
   Text1.Text = Text1.Text & fct2 & Str(fct3) & Str(fct7) & crlf

)

That factorial raised to the 2016 thus has these factors  4052160, 2024064 and 673344 times respectively.

The power-of-2 requirements for 2016 would be satisfied 4052160/5 = 810432 times; power-of-3, 2024064/2 = 1012032 times and power-of-7, 673344 times.  The limiting factor is then the power of 7: k can be as high as 673,344.

2015 has as factors 5, 13 and 31, each once.

The counts for these factors in 2015! are 502, 166 and 67 respectively, so the 31-as-a-factor will be the limiting one. When raised to the 2015, this comes to 67*2015= 135,005, for the highest allowable k.



  Posted by Charlie on 2015-02-25 10:50:12
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