Direct Proof With Odd Integers

In summary, the conversation discusses the relationship between odd integers and their divisors. It is stated that if m is an odd integer and n divides m, then n is also an odd integer. The solution involves assuming m is odd and showing that n must also be odd by contradiction.
  • #1
smiles988
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Homework Statement


If m is an odd integer and n divides m, then n is an odd integer.


Homework Equations


Odd integers can be written in the form m=2k+1.
Since n divides m, there exists an integer p such that m=np


The Attempt at a Solution


We will assume that m is an odd integer and that n divides m. We will show that n is an odd integer. Since m is an odd integer, there exists an integer k such that m=2k+1. Since n divides m, there exists an integer p such that m=np.

I don't know where to go from here to arrive at n is equal to 2*an integer +1. Do I need to use cases?
 
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  • #2
so you have m = np and m is odd. you want to show n is odd. so suppose it's even, then look what happens, you should get a contradiction.
 

1. What is a direct proof with odd integers?

A direct proof with odd integers is a mathematical proof technique that involves using the properties of odd numbers to prove a statement or theorem. This type of proof is commonly used in number theory and algebra.

2. How do you construct a direct proof with odd integers?

To construct a direct proof with odd integers, you start by assuming that the statement is true for any odd number. Then, you use mathematical properties and logical reasoning to show that the statement holds true for any odd number, thereby proving the statement for all odd integers.

3. What are some common properties of odd integers used in direct proofs?

Some common properties of odd integers used in direct proofs include:

  • The sum of two odd integers is always an even integer.
  • The product of two odd integers is always an odd integer.
  • The difference between two odd integers is always an even integer.
  • The square of any odd integer is always an odd integer.

4. Can a direct proof with odd integers be used to prove statements about even integers?

Yes, a direct proof with odd integers can be used to prove statements about even integers. This is because even integers can be written as the sum or difference of two odd integers, so the properties of odd integers can still be applied in the proof.

5. What are some real-world applications of direct proof with odd integers?

Direct proof with odd integers has many real-world applications, including in cryptography, where it is used to prove the security of encryption algorithms. It is also used in the study of prime numbers and in the development of algorithms for solving complex mathematical problems.

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