Q.310. Useful property of multiplication.

Question 310.
What is the single-digit sum of the digits in 4444^4444 ?

Answer 310.
(4444)^2 = 4444 * 4444 will have sum = (4+4+4+4) x (4+4+4+4) = 4 in single digit
(4444)^3 = 4444 * (4444)^2 will have sum = 4 * (4+4+4+4) = 1 in single digit
(4444)^4 = 4444 * (4444)^3 will have sum = 1 * (4+4+4+4) = 7 in single digit
(4444)^5 = 4444 * (4444)^4 will have sum = 7 * (4+4+4+4) = 4 in single digit
Thus, with index 2, 3, 4, 5 the single digit sum is 4, 1, 7, 4
=> with index 2, 5, 8, 11, .... 2 + (n-1)*3 = 3n-1 the sum will be 4
=> sum will be
4 for index 4442,
1 for index 4443 and will be 
7 for index 4444.
Answer : 7.

Link to YA!