## Sum of Powers of Five

Compute the units digit of $1^5+2^5+3^5+4^5+5^5+6^5+\cdots+2006^5+2007^5$.
Source: NCTM Mathematics Teacher 2008

SOLUTION 1
$1^5=1$
$1^5+2^5=33$
$1^5+2^5+3^5=276$
$1^5+\cdots+4^5=1300$
$1^5+\cdots+5^5=4425$
$1^5+\cdots+6^5=12201$
$1^5+\cdots+7^5=29008$
$1^5+\cdots+8^5=61776$
$1^5+\cdots+9^5=120825$
$1^5+\cdots+10^5=220825$
$\cdots$
$1^5+\cdots+40^5=734933200$
The units digits of $1^5+2^5+3^5+\cdots+n^5$ follows the following pattern.
If $n\equiv 0\bmod 4$, the units digits of the sum are $0,6,8,6,0$. For example,
$n=4$, units digit = $0$
$n=8$, units digit = $6$
$n=12$, units digit = $8$
$n=16$, units digit = $6$
$n=20$, units digit = $0$
If $n\equiv 1\bmod 4$, the units digits are $1,5,5,1,3$. For example,
$n=1$, units digit = $1$
$n=5$, units digit = $5$
$n=9$, units digit = $5$
$n=13$, units digit = $1$
$n=17$, units digit = $3$
If $n\equiv 2\bmod 4$, the units digits are $3,1,5,5,1$. For example,
$n=2$, units digit = $3$
$n=6$, units digit = $1$
$n=10$, units digit = $5$
$n=14$, units digit = $5$
$n=18$, units digit = $1$
If $n\equiv 3\bmod 4$, the units digits are $6,8,6,0,0$. For example,
$n=3$, units digit = $6$
$n=7$, units digit = $8$
$n=11$, units digit = $6$
$n=15$, units digit = $0$
$n=19$, units digit = $0$
Since $2007\equiv 3\bmod 4$, the units digit of the sum = $8$.

SOLUTION 2
$1^5=1$
$2^5=32$
$3^5=243$
$4^5=1024$
$5^5=3125$
$6^5=7776$
$7^5=16807$
$8^5=32768$
$9^5=59049$
$10^5=100000$
In a base-10 numbering system the units digit of $n^5$ is the same as the units digit of $n$.
Sum of units digits of $1^5+2^5+3^5+4^5+5^5+6^5+7^5+8^5+9^5+10^5= 1+2+3+4+5+6+7+8+9+0=45$
Sum of units digits of $11^5+12^5+13^5+14^5+15^5+16^5+17^5+18^5+19^5+20^5=45$
Sum of units digits of $21^5+22^5+23^5+24^5+25^5+26^5+27^5+28^5+29^5+30^5=45$
$\cdots$
Sum of units digits of $1^5+2^5+3^5+\cdots+2000^5=200\times 45=9000$
Sum of units digits of $2001^5+2002^5+2003^5+\cdots+2006^5+2007^5=1+2+3+4+5+6+7=28$
The units digit of $1^5+2^5+3^5+\cdots+2006^5+2007^5$ is $8$.

Answer: $8$