Number Theory least divisor of integer is prime number if integer is not prime

In summary, the conversation discusses a statement from a book that states the least divisor (excluding 1) of an integer a is prime if a itself is not prime. The person has tried a few examples and found them to be true, but is having trouble proving it for all cases. They eventually come to the realization that any non-prime divisor can be further divided.
  • #1
teddyayalew
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Homework Statement


The question is not really a question from a book but rather a statement that it makes : it says " Obviously the least divisor[excluding 1] of an integer a is prime if a itself is not prime." I kind of believe this statement but I'm having trouble proving the general case

Homework Equations





The Attempt at a Solution


when I take a few examples : a =8 , 2 (LD) is prime . a = 10, 2(LD) . a = 9, 3(LD) is prime a =121, 11(LD) is prime. But I'm having trouble generalizing this for all n.
 
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  • #2
any non-prime divisor can be further divided.
 
  • #3
Thank you! I feel stupid but happy I understand now.
 

1. What is Number Theory?

Number Theory is a branch of mathematics that studies the properties and relationships of integers. It is concerned with the study of numbers, particularly prime numbers, and their patterns and behaviors.

2. What is the least divisor of an integer?

The least divisor of an integer is the smallest positive integer that can divide the given integer without leaving a remainder. For example, the least divisor of 12 is 2, since 2 is the smallest integer that can divide 12 without leaving a remainder.

3. Why is the least divisor of an integer always a prime number if the integer is not prime?

This is because a prime number is defined as a number that is only divisible by 1 and itself. Therefore, if an integer is not prime, it will have at least one other divisor besides 1 and itself. And since the least divisor is the smallest divisor, it will always be a prime number.

4. How is Number Theory related to prime numbers?

Prime numbers are a fundamental concept in Number Theory. Many theorems and concepts in Number Theory revolve around prime numbers, such as the Fundamental Theorem of Arithmetic, which states that every positive integer can be uniquely expressed as a product of primes.

5. What are some practical applications of Number Theory?

Number Theory has many practical applications, including cryptography, coding theory, and computer science. It is also used in various fields of science, such as physics and chemistry, to model and solve problems. Additionally, Number Theory has connections to other areas of mathematics, such as geometry and algebra.

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