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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 Paper and Pencil solution, part (a) | Comment 1 of 5
Well, 2016 = 2^5 * 3^2 * 7.

I suspect that 7 will be the limiting factor.

Let's count what power of 7 divides into 2016!.

2016/7 = 288, so 288 numbers in 2016! are divisible by 7.
288/7 = 41 (rounding down), so 41 of the 288 are divisible by 7^2
41/7 = 5 (rounding down), so 5 of the 41 are divisible by 7^3.
288+41+5 = 334, so 2016! is divisible by 7^334.

And 2016!^2016 is divisible by 7^(334*2016) = 7^673344.

7 is the limiting factor if :
    powers of 3 in 2016! are greater than 2*334 = 668 
and powers of 2 in 2016! are greater than 5*334 = 1670

Well, that is in fact the case,
because 2016/3 = 672 are divisible by 3.  That is enough for us to disregard 3.

As for 2, 
1008 factors of 2016 are divisible by 2
 504 factors are divisible by 4
 252 are divisible by 8, 
and we need go no further, because 1008 + 504 + 252 > 1670

The final answer to the question is 673,344/1 = 673,344

  Posted by Steve Herman on 2015-02-25 10:47:33
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