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Myriad Factorial Digit (Posted on 2011-10-01) Difficulty: 3 of 5
Reading right to left, determine the 2500th digit of 10000!

*** For an extra challenge, solve this puzzle without the aid of a computer program.

No Solution Yet Submitted by K Sengupta    
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Solution In Hebrew (spoiler) | Comment 2 of 5 |
The previous post gives the 2500th digit reading from left to right, but the puzzle requires reading from right to left.  

How many zeroes are at the end of 10000!  ?

Multiples of 5 = 2000
Multiplies of 25 = 400 (each of which add one more 5)
+
Multiples of 125 = 80
Multiples of 625 = 16
Multiples of 3125 = 3,

so the last 2499 digits are zeroes.

10! = 10*9*8*7*6*5*4*3*2*1 = 3628800,
so I think the final answer is 8^1000 (mod 10).
= 4^500 (mod 10)   (because 8*8 = 64)
= 4^100 (mod 10)   (because 4*4*4*4*4 mod 10 = 4)
= 4^20 (mod 10)
= 4^4 (mod 10)
= 6

Unless, of course, I've made a mistake, which has been known to happen




  Posted by Steve Herman on 2011-10-01 22:08:29
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