Prove That an Expression is a Multiple of 10

Date: 12/19/2002 at 23:48:40
From: Ritchie
Subject: Prove that an expression is a multiple of 10

If a and b are positive integers, prove that (a^5)*(b) - (a)*(b^5) is 
a multiple of 10.

I can't find the connection between being a multiple of 10 and the 
expression given. I thought about factoring the expression into this 
form: ab(a+b)(a-b)(a^2 + b^2), but I can't do anything with that.
I tried factoring and rearranging the expression many different ways 
but couldn't find the connection.

Date: 12/20/2002 at 04:26:43
From: Doctor Mike
Subject: Re: Prove that an expression is a multiple of 10
   
Hello Ritchie,  
  
What an interesting problem! My first reaction was to doubt the truth 
of the theorem, but it really IS true. I had to stretch my grey matter 
a bit to find out why, though. But that's okay. Stretching is good.

There may be many creative ways to prove this, but here's an outline 
of my approach. My first step was to look at the special case where 
b=1. Then the theorem says a^5 - a is a multiple of ten. There was 
considerable doubt in my mind as to whether even this is true, but I 
looked at a few examples.  2^5-2 is 30.  3^5-3=240.  8^5-8=32760. You 
should make a table with 3 columns. The first column is N, the second 
is N^5, and the third is N^5-N. Fill in this table for N = 1 up to 
N = 9. Notice that the units digit of all the numbers in column three 
is zero.

Now we know, at least for 1-digit numbers N, that N^5-N is a multiple 
of ten, and we also know the related fact that N^5 has the same units 
digit as N does. Think about it.

The next step of the proof is to extend this result to all positive 
integers N, not just the ones from 1 to 9.  Just concentrate on what 
happens with the units digit when you do the multiplications for N*N*
N*N*N = N^5.  Do it for something like 13^5 and see how it works.

A special simpler result used to prove a more general result is called 
a Lemma. That's what this is. The fact that N^5-N is a multiple of ten 
will help us prove that for (a^5)*(b) - (a)*(b^5). How? We know from
the Lemma that a^5 - a is a multiple of ten, and b^5 - b is also some 
multiple of ten. They might not be the same multiple, so let's use the 
notation of a^5 - a = (10)(K) and also b^5 - b = (10)(Q) where K and Q 
are some integers. Then a^5 = a + (10)(K) and b^5 = b + (10)(Q).

Now substitute these expressions for a^5 and b^5 into your original 
expression (a^5)*(b) - (a)*(b^5) and then simplify. You should be able 
to factor out a ten, which shows that this expression is a multiple of 
ten.

You should go back and actually do all the things that I have 
described in my proof outline, so you completely understand the whys 
and wherefores of each step.

Good luck.  And thanks for sending in such an interesting problem.
- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/