Prime number dividing fractions.

In summary, the conversation discusses proving that A/B is divisible by a prime number p and its relation to Fermat's little theorem. The solution involves decomposing A and B into prime factors and realizing that p remains a factor in A/B because B does not contain p. The fundamental theorem of arithmetic is also mentioned as a key concept in solving this problem.
  • #1
Jolb
419
29
Let p be a prime number.
Let A be an integer divisible by p but B be an integer not be divisible by p.
Let A/B be an integer.

How do I prove that A/B is divisible by p?


This sounds like a simple question but I just can't get it. I'm doing it in relation to proving Fermat's little theorem. (a^p = a mod p for all integers a) I'm trying to understand why the binomial coefficients in the binomial expansion of (1+a)^n are all divisible by p (=0 mod p) for all the terms with powers [1, p-1].
 
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  • #2
Decompose A and B into prime factors. Since A/B is an integer all the B factors cancel A factors. However B did not contain p, so p remains a factor in A/B.
 
  • #3
Thank you, mathman! That instantly resolved my question (and I was banging my head against it for like an hour)!

Damn that fundamental theorem of arithmetic!
 

FAQ: Prime number dividing fractions.

1. What are prime numbers?

Prime numbers are positive integers that are only divisible by 1 and themselves. Some examples of prime numbers are 2, 3, 5, 7, and 11.

2. How do you determine if a number is a prime number?

A number is considered a prime number if it is only divisible by 1 and itself. To determine if a number is prime, you can try dividing it by all the numbers from 2 to its square root. If the number is only divisible by 1 and itself, then it is a prime number.

3. How are prime numbers used in dividing fractions?

Prime numbers are used in dividing fractions by finding the greatest common divisor (GCD) between the numerator and denominator. The GCD is the largest prime number that is a factor of both the numerator and denominator. This can help simplify the fraction and make it easier to divide.

4. Can prime numbers be used to divide any fraction?

Yes, prime numbers can be used to divide any fraction. However, not all fractions can be simplified using prime numbers. Some fractions may have a GCD that is not a prime number, and in that case, the fraction cannot be simplified any further.

5. Why are prime numbers important in mathematics?

Prime numbers are important because they are the building blocks of all other numbers. Every positive integer can be broken down into a unique combination of prime numbers. Additionally, prime numbers have many real-world applications in fields such as cryptography, computer science, and number theory.

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