Question
Wed March 26, 2014 By: Mohit Kumar

what will be the unit digit of (3^2013)

Expert Reply
Wed March 26, 2014
Let us consider the powers of 3
3 to the power of 1 equals 3 3 squared equals 9 3 cubed equals 27 3 to the power of 4 equals 81 3 to the power of 5 equals 243 3 to the power of 6 equals 729 . . . . .
 
Let us now consider only the unit digits of powers of 3.
 
T h u s comma space w e space h a v e 3 to the power of 1 space h a s space u n i t space d i g i t space 3 3 squared space h a s space u n i t space d i g i t space 9 3 cubed space h a s space u n i t space d i g i t space 7 3 to the power of 4 space h a s space u n i t space d i g i t space 1 3 to the power of 5 space h a s space u n i t space d i g i t space 3 3 to the power of 6 space h a s space u n i t space d i g i t space 9 3 to the power of 7 space h a s space u n i t space d i g i t space 7 3 to the power of 8 space h a s space u n i t space d i g i t space 1 3 to the power of 9 space h a s space u n i t space d i g i t space 3 3 to the power of 10 space h a s space u n i t space d i g i t space 9 . . . . .
Thus there is a pattern in the unit digit of powers of 3.
 
The unit digits are appearing in cylces of 3, 9, 7, and 1.
 
Therefore, if the power is divided by 4, and the remainder is 1, the unit digit is 3.
 
And, if the power is divided by 4, and the remainder is 2, the unit digit is 9.
 
And, if the power is divided by 4, and the remainder is 3, the unit digit is 7.
 
And, if the power is divided by 4, and the remainder is 0, the unit digit is 1.
 
So let us divide the power 2013 by 4.
 
Thus, 2013 equals 503 cross times 4 plus 1
since the remainder is 1, we have the unit digit as 3.
Home Work Help