To find the minimum possible value of 'n' such that the number 677 has exactly 5 digits when converted into base 'n', we need to determine the value of 'n' that is greater than the largest digit in the base 'n' representation of 677.
The largest digit in the base 'n' representation of 677 will be one less than 'n' if 'n' is greater than or equal to 10. This is because the digits in base 'n' range from 0 to 'n-1'.
In this case, we are looking for 'n' that is greater than the largest digit, which is 6 (since 677 has 6 as its largest digit in decimal system). Therefore, the minimum possible value of 'n' is 7.
However, in the given options, the closest value is 4. So, the correct answer is option C) 4.