What are you trying to prove?dgamma3 said:Homework Statement
I know you could prove this by stating every integer is either 3m, 3m+1 or 3m+2. However I am trying to prove this just using either even numbers or odd numbers.
so for example, when I try:
(2x+1)^2
= 4x^2 + 4x + 1 - expand
= 3x^2 + x^2 + 3x + x + 1 - group like terms
= 3x^2 + 3x + x^2 + x + 1
= 3*(x^2 + x) + x(x+1) + 1
x(x+1) + 1 is an odd number
so
= 3*(x^2 + x) + 2p + 1
thats as far as I can go.
Prove that the square of any integer, when divided by 3, does what?Prove that the square of any integer, when divided by 3. only by odd and even.
The statement being proven is that the square of any integer, when divided by 3, will result in either an odd or even number.
This statement is significant because it provides a rule or pattern for determining whether a number is odd or even, without having to perform the actual division calculation.
This statement can be proven using the properties of even and odd numbers, as well as the properties of division.
The steps to prove this statement are as follows: 1) Begin with an arbitrary integer, n. 2) Square n to get n^2. 3) Divide n^2 by 3. 4) Use the properties of even and odd numbers to show that the resulting number is either even or odd. 5) Since n was arbitrary, this pattern applies to all integers.
Using the properties of even and odd numbers in the proof helps to show that the resulting number after division by 3 will always be either even or odd, regardless of the original integer used. This strengthens the validity of the statement being proven.