Square of Odd Integers & Justifying "If P2 Is Even, Then P Is Even

[winsome]

show that the square of any odd integer is odd, use this fact to justify the statement "if p2
is even , then p is also even

 

[micromass]

Well, what did you try already? If we know where you're stuck, then we'll know where to help...

 

[gauss^2]

A good first step is noticing that

...
-5 = 2*(-3) + 1
-3 = 2*(-2) + 1
-1 = 2*(-1) + 1
1 = 2*0 + 1
3 = 2*1 + 1
5 = 2*2 + 1
...

 

[Amergin]

(2x+1) is odd. (2x+1)^2=4x^2+4x+1 but 4x^2+4x is even and even +1 gives odd.
So (2x+1)^2 is odd

 

[blue_raver22]

."

I would like to start by clarifying the definitions of "odd integer" and "even integer." An odd integer is any number that is not divisible by 2, while an even integer is any number that is divisible by 2. With this in mind, we can say that the square of an odd integer, let's call it p, would be p^2.

To prove that the square of any odd integer is also odd, we can use the fact that any odd integer can be expressed as 2n+1, where n is any integer. Substituting this into p^2, we get:

p^2 = (2n+1)^2 = 4n^2 + 4n + 1

We can see that the resulting expression has a remainder of 1 when divided by 2, meaning that p^2 is not divisible by 2. This confirms that the square of any odd integer is indeed odd.

Now, let's look at the statement "if p^2 is even, then p is also even." If p^2 is even, then it means that it is divisible by 2. Using the same expression as before, we get:

p^2 = (2n)^2 = 4n^2

Since p^2 is divisible by 2, it follows that p is also divisible by 2. And as we established earlier, any number that is divisible by 2 is an even integer. Therefore, we can justify the statement "if p^2 is even, then p is also even."

In conclusion, the square of any odd integer is odd and the statement "if p^2 is even, then p is also even" can be justified by using this fact. This shows the importance of understanding the properties and characteristics of numbers in mathematics, which allows us to make logical and accurate statements and conclusions.