Proving: If a is Even, Then 4 Divides a

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In summary, we are given that a and b are natural numbers such that a^2 = b^3. We are asked to prove that if a is even, then 4 divides a. By definition, an even number can be represented as 2n, where n is an integer. We can also represent b as 2m, where m is another integer. Substituting these values into a^2 = b^3, we get (2n)^2 = (2m)^3, which simplifies to 4n^2 = 8m^3. By dividing both sides by n, we get 4n = 8m^3/n. Since 8m^3 is divisible by
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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.
 
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  • #2
Try to show: if 2 divides b3, then 8 divides b3.
 
  • #3
What you have done is pretty good up to here...

2n = 4m^3/n

You have already shown that

[tex] 4n^{2} = 8m^3[/tex]

[tex] n^{2} = 2m^3[/tex]

This imples n is even, do you see why ?

If n is even then 2n is definitely divisible by 4.
 
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  • #4
╔(σ_σ)╝ said:
What you have done is pretty good up to here...



You have already shown that

[tex] 4n^{2} = 8m^3[/tex]

[tex] n^{2} = 2m^3[/tex]

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.
 

1. What does it mean for a number to be even?

Being even means that a number is divisible by 2, or can be divided into two equal groups without any remainder.

2. How do you prove that a number is even?

A number is even if it can be written in the form 2n, where n is any integer. This can be proven by dividing the number by 2 and checking for a remainder of 0.

3. What does it mean for 4 to divide a number?

When 4 divides a number, it means that the number can be split into four equal groups without any remainder. In other words, the number is divisible by 4.

4. How do you prove that if a is even, then 4 divides a?

To prove this statement, you can use the definition of an even number and the definition of dividing a number by another number. You can also use mathematical induction to show that the statement holds true for all even numbers.

5. Can a number be even and not divisible by 4?

Yes, a number can be even and not divisible by 4. For example, 6 is an even number but it is not divisible by 4 because it leaves a remainder of 2 when divided by 4.

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