+0

# units digit of 3^2019 in base 16

0
93
2

Find the units digit of 3^2019 in base 16.

May 13, 2022

#1
0

3^2019 mod 10^10==4B5998A3F1C1AAA3 - these are the 10 "digits" in base 16

May 14, 2022
#2
+117821
+1

$$3^0=1 \;\;(base10)\;\;\equiv 1 \;\;(base16)\;\; \\ 3^1=3 \;\;(base10)\;\;\equiv 3 \;\;(base16)\;\; \\ 3^2= \;\;(base10)\;\;\equiv 9 \;\;(base16)\;\; \\ 3^3=27 \;\;(base10)\;\;\equiv B \;\;(base16)\;\; \\ 3^4=81 \;\;(base10)\;\;\equiv 51 \;\;(base16)\;\; \\$$

The pattern for the last digit is now set.

$$3^{0+4n} \;\;ends \;\;in\;\; 1 \;\;(base16)\;\; \\ 3^{1+4n} \;\;ends \;\;in\;\; 3 \;\;(base16)\;\; \\ 3^{2+4n}\;\;\;\;ends \;\;in\;\; 9 \;\;(base16)\;\; \\ 3^{3+4n} \;\;\;\;ends \;\;in\;\;B \;\;(base16)\;\; \\$$

2019 = 4*504+3

So

$$3^{2019}\quad \text{has a unit digit of WHAT in base 16.}$$