Proving: If a is Even, Then 4 Divides a

[number0]

Homework Statement

Let a and b be a natural numbers such that a² = b³. Prove the following proposition:
If a is even, then 4 divides a.

Homework Equations

Definition: A nonzero integer m divides an integer n provided that there is an integer q such that n = m * q.

Definition: A even number m can be represented by the relationship m = 2 * n where n is an integer.

The Attempt at a Solution

Let a = 2n where b is any integer. Let b = 2m where m is any integer (from another theorem, the cube of any even is even).

a^2 = b^3
(2n)(2n) = 8m^3
2n = 4m^3/n

I am not even sure if I did all of the above correctly; but, this is as far as I got.

 

[micromass]

Try to show: if 2 divides b³, then 8 divides b³.

 

[╔(σ_σ)╝]

What you have done is pretty good up to here...

2n = 4m^3/n

You have already shown that

$$4n^{2} = 8m^3$$

$$n^{2} = 2m^3$$

This imples n is even, do you see why ?

If n is even then 2n is definitely divisible by 4.

 

[number0]

What you have done is pretty good up to here...



You have already shown that

$$4n^{2} = 8m^3$$

$$n^{2} = 2m^3$$

This imples n is even, do you see why ?

If n is even then 2n is definitely divisible by 4.

Wow... thank you! I finally see how it is done.