Amazingly, the answer can be found with a normal calculator. Let X be 4444^4444, and Y = sod(sod(sod(X))).

Let's calculate Y's remainder upon division by 9. It is a well-known fact

that n == sod(n) (mod 9) for all positive numbers. Therefore, Y == X (mod 9). 4444 == 7 (mod 9), so Y == X == 7^4444 (mod 9). We also discover that 7^3 == 1 (mod 9), so 7^4444 = 7 * (7^3)^1481 == 7 (mod 9). Therefore,

Y == 7 (mod 9) (*)

Furthermore,

X < 10000^4444 = (10^4)^4444 = 10^(4*4444) = 10^17776.

Of all numbers less than 10^17776, the one with the highest total digital sum is 10^17776-1, i.e. 17776 9's. Therefore, sod(X) <= 9 * 17776 =

159984. Similarly, sod(sod(X)) <= sod(99999) = 45, and we do it again to get
Y <= sod(39) = 12 (**)

The only positive number that satisfies both (*) and (**) is 7, so sod(sod(sod(4444^4444))) = 7.

Matthew Daly mwdaly@kodak.com

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