Determine the unit digits of 6661984SolutionSince we are int

Determine the unit digits of 666^1984

Solution

Since we are interested only in the units digits we can ignore the tens and hundreds place digits of the number 666 so we need only find the unit digits of 6^1984

6^2=36, unit digit is 6

6^3=36*6=216, unit digit is 6. So we see that the unit digit of all powers of x remains 6. Hence unit digit of 666^1984 is 6

We can prove this by induction

Base case:

n=1, 6^1 has unit digit 1

Let it be true for n>=1

We prove it for n+1

6^(n+1)=6*6^n

Since 6^n has unit digit 6 as we assumed so 6*6^n has unit digit 6. Hence for all values of n=1,2,3,...

6^n has unit digit 6

Hence, 666^n has unit digit 6 for all values of n=1,2,3,...

Hence, 666^1984 has unit digit 6.