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Non - Zero Digit II (Posted on 2012-02-05) Difficulty: 4 of 5
Reading from left to right, what is the last non - zero digit in ((20!)!)!? (That is, the factorial of factorial of 20 factorial).

No Solution Yet Submitted by K Sengupta    
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Some Thoughts I agree with Jer | Comment 4 of 7 |
I do not know the last non-zero digit of ((20!)!)!, except it is one of the following: 2, 4, 6, or 8.

As Jer pointed out, the pattern appears to be reflected in base-5, and the pattern seems to "shift/change" at each 5nth power.

There are four base patterns, with a base pattern being the last non-zero digit of a sequence of {n!, (n+1)!, (n+2)!, (n+3)!, (n+4)!} such that n is a multiple of 5.
The four base patterns are:
A: {2, 2, 4, 2, 8},
B: {4, 4, 8, 4, 6},
C: {6, 6, 2, 6, 4}, and
D: {8, 8, 6, 8, 2}.

The next level, representing a sequence {5n!, (5n+1)!, (5n+2)!, (5n+3)!, (5n+4)!}, is more complex in that each pattern is a series of the base patterns:
E: {A, B, C, C, D}
F: {B, D, A, A, C}
G: {C, A, D, D, B}
H: {D, C, B, B, A}

Again the complexity increases as the sequence is further extended at the next power {25n!, (25n+1)!, (25n+2)!, (25n+3)!, (25n+4)!} where one can find one of the four following as the pattern:
I: {E, H, F, H, H}
J: {F, G, H, G, G}
K: {G, F, E, F, F}
L: {H, E, G, E, E}.

As ((20!)!)! is an exhaustively huge number with (20!)! even too large for the BIG NUMBER CALCULATOR to present a number (the largest factorial that can be computed using the  
calculator is 999!), it is even difficult to determine what order of 5 to consider. 
  Posted by Dej Mar on 2012-02-06 16:42:13
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