The number of digits of the nth power of a positive integer The number of digits of positive integer power is regular. When the number of digits of positive integer a is 0,1,5,6, the number of digits of a ^ n is still 0,1,5,6; when the number of occurrences of a is 4,9, the number of digits of power will appear repeatedly every time the exponent of a increases by 2; when the number of digits of a is 2,3,7,8, the number of digits of power will appear repeatedly every time the exponent increases by 4 If the single digit of a ^ k is a, then the last digit of a ^ 4m + k is also a (k is a positive integer, M is a non negative integer) Example 1 find the number of 2003 ^ 2005 2003^2005=2003^4×501+1. Because the single digit of 2003 ^ 1 is the same as that of 3 ^ 1, the single digit of 2003 ^ 2005 is 3 Example 2 find the integer x satisfying the equation x ^ 5 = 656356768 ∵ 10 ^ 5 is 6 digits. 100 ^ 5 = 10 ^ 10 is 11 digits Because x ^ 5 is 9 digits. So 10

one
2^2008 = 2^(501*4+4)
The number of single digits is the same as that of 2 ^ 4
2^4 = 16
SO 2 ^ 2008 is 6
two
10^5 < 1419857