1989 maps itself

Number Theory Level 2

What is the last digit of $1989^{1989}$ ?

2 0 6 3 9 1 7 8

2 solutions

Smruti Charkha
Jun 19, 2015

Note: here ^ means power. I use the logic,

consider,

9^1 = 9

9^2 = 81

9^3 = 729

9^4 = 6561

and so on.

Thus for odd power unit digit = 9 and

for even power unit digit = 1

Hence the answer is 9.

Félix de Montemar
Apr 9, 2015

${ 1989 }^{ 1989 }\equiv a\quad (mod\quad 10)$ ${ 1989 }^{ 1989 }={ 1989 }^{ 1988 }·9={ \left( 10·199-1 \right) }^{ 1988 }·9\equiv \left( \dot { 10 } +1 \right) ·9\quad (mod\quad 10)\left( 10·199-1 \right) \equiv 9\quad (mod\quad 10)$ , so it ends in 9

Or you can simply use some basic modular arithmetic noting that $1989$ is odd.

$\begin{aligned}1989^{1989}=(1990-1)^{1989}\equiv (0-1)^{1989}&\equiv (-1)^{1989}\\&\equiv -1\\&\equiv -1+10\equiv 9\pmod{10}\end{aligned}$

Prasun Biswas - 5 years, 12 months ago