
Why is the last digit of n^5 equal to the last digit of n? [video] - iamzlatan
https://www.youtube.com/watch?v=ZQUTV9or98s
======
contravariant
A somewhat more general proof involves the use of the Chinese remainder
theorem and the fact that 4 is divisible by the order of the multiplicative
group of both integers mod 2 and mod 5. Therefore x^5 = x mod 2 and x^5 = x
mod 5 for all x, and as a consequence x^5 = x mod 10 for all x.

This construction seems to fail if the base isn't square free. In particular
in base 4 the number 2 has as powers 2, 10, 20, 100 etc.

~~~
knappa
That's not true for x=2 : 2^4 = 6 mod 10 Also, 4 does not divide the order of
the multiplicative group of Z/2Z. i.e. 4 does not divide 1. (Also, the claim
was that x^5 = x mod 10 not x^5=1, but I assume that's a typo.)

~~~
contravariant
Hmm yes I suppose I was a bit sloppy, although I got the gist of it right
somehow. I guess the problem here is that x^4 = 1 mod 2 is false if x = 0 mod
2, but x^5 = x mod 2 does hold for all numbers.

I've fixed the problems you mentioned, thanks.

------
panda88888
At work so I haven’t watched the video yet, but my quick rationale is,

Let m = abs(n) It follows n = sign(n) * m We can work with m since sign(n)
doesn’t affect the ones digit of n^5. More formally, n^5 = sign(n)^5 *
abs(n)^5.

Let m = 10 * b + a, where b = floor(m/10) a = m % 10

Essentially we extract the ones digit as a.

m^5 = (10 * b + a)^5 If we expand the terms, only a^5 affects the ones digit.
The remaining has at least one 10 as factor.

Quick python script will verify a^5 has a as the ones digit for 0...9.

    
    
      for a in range(10):
          print(pow(a, 5) == a) 
    

Edit: fixed formatting

~~~
contravariant
There are some slight problems with your proof, firstly your proof doesn't
really use the fact that you're raising a number to the 5th power, except for
the last part and secondly the last part (as it is currently written) will
print "false" for all numbers other than 1.

Also I don't think you need to bother with the sign of the number. Python is
one of the few languages that took the sensible approach and made statements
like (a % 10 = (a + 10) % 10) true for all a.

------
blt
good example of an inductive proof, would be helpful for people learning about
inductive proofs.

