Nestor was doing the work of his math class about three days but he is tired of make operations a lot and he should deliver his task tomorrow. His math’s teacher gives two numbers a and b. The problem consist in find the last digit of the potency of base a and index b. Help Nestor with his problem. You are given two integer numbers: the base a (0 <= a <= 20) and the index b (0 <= b <= 2,147,483,000). You have to find the last digit of a^b.
Problem: Sphere Online Judge (SPOJ) – Problem LASTDIG
Solution: I don’t really have an background in number theory, so I decided to approach this problem empirically. My main source of information was this small Python program:
for i in xrange(1, 20):
print i, str(base ** i)[-1:]
I checked the last digit for each base (0..20) and looked for patterns. Happily, I found them. Here’s a snippet:
x > 0
0 ^ x => 1
1 ^ x => 1
2 ^ x => 2^1 -> 2
2^2 -> 4
2^3 -> 8
2^4 -> 6
3 ^ x => 3^1 -> 3
3^2 -> 9
3^3 -> 7
3^4 -> 1
4 ^ x => 4^1 -> 4
4^2 -> 6
5 ^ x => 5
The rest is straight forward and thanks to the 700B source code size limitations, the source code is pretty unreadable. ;)