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

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About mvtrinh

Retired high school math teacher.
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