intro to proofs question

Question

using direct proof (axioms, definitions, etc)
for all integers n and m, if n-m is even then n cubed-m cubed is even.
I believe this is true I just don't know how to prove it.

fluke · Answer

n-m=even 
 n^3-m^3=(n-m)(n^2+nm+m^2)=even*(n^2+nm+m^2)=even*integer=even

Even multiplied by any integer is even.

helpmewithmath · Answer

is that an axiom? Cause I haven't learned that one yet...

fluke · Answer

I believe it should be a theorem because it has a proof. Either way, it is believed to be true.

Even*odd = even
e.g.
2*3=6

Even*even = even
2*2=4

Please go through the number properties:
 https://gmatclub.com/forum/math-number-theory-88376.html

helpmewithmath · Answer

thanks so much you've been a huge help!!!

KarishmaB · Answer

My suggestion would be not worry about theorems/axioms. Rather, try and use logic to convince yourself whether something is true.

n - m is even means that either both n and m are even or both are odd
4-2 = 2
3-1 = 2
4-3 = 1 (If one is even and other is odd, then the result will always be odd)

Again, you don't need to memorize this. Think of an even number as 2a since an even number is a multiple of 2. Think of an odd number as (2a+1) since it is 1 more than a multiple of 2.
When you subtract 2 even numbers: 2a - 2b = 2(a - b), that is an even number
When you subtract 2 odd numbers: (2a+1) - (2b+1) = 2(a-b), again an even number
When you subtract an odd and an even number: (2a+1) - 2b = 2(a-b) + 1, which is an odd number
Just imagine odd numbers with an extra 1. When you subtract an even number from an odd number, the extra 1 remains.
But when you subtract an even number from an even number, there is no extra 1 anyway and when you subtract an odd number from an odd number, both have extra 1s so they get canceled out...
Thinking this way will make it quick and intuitive.

Coming back to the original question, 
n - m = even means that either both n and m are even or both are odd.
Now, think if n is even, wouldn't  n^3  be even too? Yes it will because since n has 2 as a factor,  n^3  will also have 2 as a factor. In fact it will have three 2s in it. Say, if n = 2a,  n^3 = 2a*2a*2a 
Similarly, if m is even,  m^3  is definitely even too.

n^3 - m^3  = Even - Even = Even

On the same lines, 
if n is odd,  n^3  is definitely odd too. Since n does not have 2 as a factor, no matter how many n's you multiply with each other, you will never have 2 as a factor. Say if n = 3,  n^3 = 3*3*3 . It will not have a 2.
Similarly, if m is odd,  m^3  is odd too.
 n^3 - m^3 = Odd - Odd = Even

And once you understand this well, all this will flash through your mind in a matter of seconds...

intro to proofs question

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