Mindblowing SOD!

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>>> num=0
>>> def SOD(num):
    return sum(map(int,str(num)))

>>> SOD(SOD(SOD(4444**4444)))
>>> 7

Thus the answer would be 7

The answer is the congruence mod 9 or the so called digital root. But first we need to check the proper bounds. Let X = 4444^4444 X= 4444^4444 has fewer than 4444 4 = 17776 digits. Therefore SOD(X) is less than 17776 9 = 159984 (but really is much smaller) Then SOD(X) has at most 6 digits and the first two can't add more than 1+5. Therefore SOD(SOD( X) ) is less than 1+ 5+ 9 4=42 And so SOD(SOD(SOD(X))) is less than 18. Now, 4444 mod 9 = 7 and 7^3 = 1 mod 9 So 4444^4444 = 7^4444 = 7 ( 7^4443) = 7 (7^3)^1481 = 7(1^1481) = 7 1 = 7 all calculations done mod 9.

Check out this problem . If you can do the current problem, you can also do my problem. If not, you can find the solution there.