Last digit of a sum of numbers with power.
Last digit of a sum of any number of numbers is equal to last digit of sum of last digits of all the numbers.
So to find the last digit in this case first we should consider how to find the last digit of any power of a number.
After calculating the last digit of each numbers with power we can add them and get our required last digit.
So lets consider a problem :
Q. Find the last digit of 56747^987 + 987^56747 + 34545^23 ?
Last digit of 56747^987 = Lat digit of 7^987.
Now cyclicity of 7 is 4.
So we divide 987 by 4 and consider the remainder part.
987 /4 gives 3 as remainder.
So last digit of 56747^987 = last digit of 7^3 = 3.
Last digit of 987^56747 = Last digit of 7^56747
Now again cyclicity of 7 is 4
So we divide 56747 by 4 and consider the remainder part
56747/4 gives 3 as remainder.
So last digit of 987^56747 = last digit of 7^3 = 3.
Last digit of 34545^23 = Last digit of 5^23
Cyclicity of 5 is 2
So we divide 56747 by 2 and consider the remainder part
56747/2 gives 1 as remainder.
So last digit of 34545^23 = last digit of 5^1 = 5.
Hence required last digit = 3 + 3 + 5 ~ 1.