Proving Statement: If m^2 is 4k+3, Then m is 4k+3

In summary, the statement given is: if m^2 is of the form 4k+3, then m is of the form 4k+3. The conversation discusses using the method of contraposition to prove this statement. It is explained that all integers m fall into one of the three forms 4k, 4k+1, 4k+2, or 4k+3. The only way for both m^2 and m to be of the form 4k+3 is if m=1. The conversation also mentions that this problem has been posted and answered multiple times.
  • #1
CollectiveRocker
137
0
I'm given the statement: if m^2 is of the form 4k+3, then m is of the form 4k+3. I don't even know how to begin proving this. I'm guessing by contraposition.
 
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  • #2
You put this already under general math
 
  • #3
good guess. if m is not of form 4k+3 thern m is of form 4k, 4k+1, or 4k+2. see what that gives.
 
  • #4
Please explain what you mean.
 
  • #5
all integers m are of one of the three forms 4k, 4k+1, 4k+2, or 4k+3, since when you divide an integer by 4, you get a remainder which is either 0,1,2, or 3.
 
  • #6
The only way that m^2 = 4k+3 AND m = 4k+3 is if m=1

Is this what you meant?
 
  • #7

1. What is the statement "If m^2 is 4k+3, Then m is 4k+3" trying to prove?

The statement is trying to prove that if the square of any integer, m, is of the form 4k+3 (where k is any integer), then m itself must also be of the form 4k+3.

2. Why is this statement important?

This statement is important because it is a fundamental property of integers and helps us understand the behavior of numbers in the form of 4k+3. It also has applications in various fields of mathematics, such as number theory and algebra.

3. How can this statement be proven?

This statement can be proven using proof by contradiction. Assuming that the statement is false, we can show that it leads to a contradiction, thus proving that the statement is true.

4. Can you provide an example to illustrate this statement?

Sure, let's take m = 5. The square of 5 is 25, which is of the form 4k+1. This contradicts the statement, so we can conclude that the statement is true.

5. Is this statement applicable to all integers?

Yes, this statement is applicable to all integers. It is a universal truth and can be applied to any integer in the form of 4k+3.

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