Congruency and Divisibility of Odd in Z

In summary: I understand that ##d | n+2 -n #### n+2 -n = dq, ## where q ## \in Z##Since, 2 = dq, so d and q could be 1 or 2.
  • #1
knowLittle
312
3

Homework Statement


Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2.

The Attempt at a Solution


I don't think I understand the question.

The few notes I have state d| (n+2 )- n

This resembles n+2 ##\equiv## n mod d , but I don't see the connection.
The congruency means that n+2 and n share the same remainder.
Any help?
 
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  • #2
knowLittle said:

Homework Statement


Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2.

The Attempt at a Solution


I don't think I understand the question.

The few notes I have state d| (n+2 )- n

This resembles n+2 ##\equiv## n mod d , but I don't see the connection.
The congruency means that n+2 and n share the same remainder.
Any help?

The problem doesn't have anything to do with congruency. If d is a common divisor then d divides n and d divides n+2, so it follows that d must divide (n+2)-n. Don't you agree? Now use that to solve the problem.
 
  • #3
Thank you. I agree.
I understand that
##d | n+2 -n ##

## n+2 -n = dq, ## where q ## \in Z##
Since, 2 = dq, so d and q could be 1 or 2.
 
  • #4
knowLittle said:
Thank you. I agree.
I understand that
##d | n+2 -n ##

## n+2 -n = dq, ## where q ## \in Z##
Since, 2 = dq, so d and q could be 1 or 2.

Correct, that d must be 1 or 2. Now can you prove the original problem?
 
  • #5
I thought that I already did.
d |n and d| n+2
n =dk and n+2 = dp, k and p are in Zn is odd, so n =2a+1 for some a in Z.

dk +2 =dp
2a+1 +2 = dp
2(a+1) + 1 = dp

I think this means that I am confused.
 
  • #6
knowLittle said:
I thought that I already did.
d |n and d| n+2
n =dk and n+2 = dp, k and p are in Z


n is odd, so n =2a+1 for some a in Z.

dk +2 =dp
2a+1 +2 = dp
2(a+1) + 1 = dp

I think this means that I am confused.

Yeah, it does look like you have lost your way. It's much simpler than that. You know d=1 or 2. You are given that n is odd. d divides n. Can d be 2?
 
  • #7
An even number can never divide an odd number. 'd' must be 1. :>

Thanks.
 

Related to Congruency and Divisibility of Odd in Z

1. What is Congruency in Z and how is it determined?

Congruency in Z refers to the property of two integers being in the same equivalence class. In other words, two integers are congruent if they have the same remainder when divided by a given number. This can be determined using the modulus operator, which gives the remainder of a division operation.

2. How do you determine if a number is divisible by another number in Z?

A number is divisible by another number in Z if the remainder of the division is equal to zero. This can be determined using the modulus operator, which gives the remainder of a division operation. If the remainder is zero, then the number is divisible by the given number.

3. What is the relationship between congruency and divisibility in Z?

Congruency and divisibility are closely related in Z. If two numbers are congruent modulo a given number, then they are also divisible by that number. On the other hand, if two numbers are divisible by a given number, then they are congruent modulo that number. This relationship is important in number theory and has applications in cryptography and coding theory.

4. How can congruency and divisibility be used to solve problems in number theory?

Congruency and divisibility are fundamental concepts in number theory and are used to solve problems related to prime numbers, modular arithmetic, and Diophantine equations. These concepts help to identify patterns, relationships, and properties of integers, which can then be used to solve more complex problems in mathematics.

5. Can congruency and divisibility be extended to other number systems besides Z?

Yes, congruency and divisibility can be extended to other number systems, such as rational numbers, real numbers, and complex numbers. However, the definitions and properties may differ from those in Z. In general, congruency and divisibility are important concepts in abstract algebra and have applications in various areas of mathematics and computer science.

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