once that digit mod 5 is the same as your starting digit, you are done

That assumes there is at least one entry for each minute... What if the log looked like:

2:00:00 all is well
2:01:00 all is well
2:02:00 all is well
2:03:00 all is well
2:04:00 oops, reboot
2:21:00 all is well again
[download]

I think looking at more than just that one digit is wise.

-Blake