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:

1 2 | 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:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | 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. ;)