Number Theory Problem: Proving (a,b)=1 if a|c and b|c

In summary, we are given that a,b,c belong to Z with (a,b)=1. We need to prove that if a|c and b|c, then ab|c. Using the prime factorization of e, we can show that c=acx+bcy=abrx+basy, where ax+by=1. Therefore, ab|c.
  • #1
yeland404
23
0

Homework Statement



a,b,c belong to Z with (a,b)=1. Prove that if a|c and b|c, then ab|c

Homework Equations


let a1,a2...an, c belong to Zwith a1...an pairwise relatively prime, prove if ai|c for each i, then a1a2...an|c


The Attempt at a Solution



if a|c, then c=ea, b|c, then c=fb, then which the next step and how it relates with (a,b)=1
 
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  • #2
(a,b)=1, thus consider the prime factorization of e.
 
  • #3
There exists integers x, y such that ax+by=1. Therefore c=acx+bcy=abrx+basy.
 

1. What is number theory?

Number theory is a branch of mathematics that deals with the properties and relationships of integers. It is concerned with the study of numbers, their patterns, and their interactions with each other.

2. What is a number theory problem?

A number theory problem is a question or challenge that requires the application of number theory concepts and techniques to find a solution. These problems can range from simple arithmetic puzzles to complex mathematical conjectures.

3. What are some common types of number theory problems?

Some common types of number theory problems include prime number problems, divisibility problems, modular arithmetic problems, and Diophantine equations. These types of problems often involve finding patterns or relationships between numbers.

4. How is number theory used in real-world applications?

Number theory has many practical applications, including cryptography, coding theory, and data encryption. It is also used in fields such as physics, computer science, and engineering to solve problems and develop new technologies.

5. What are some strategies for solving number theory problems?

Some strategies for solving number theory problems include breaking down the problem into smaller parts, using patterns and relationships between numbers, and applying known theorems and formulas. It is also helpful to approach the problem from different angles and to use trial and error to test solutions.

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