I saw on the internet the following Proof of Euclids lemma which states that if a prime number divides the product of two numbers then it must divide at least one of them.
Since p divides bc, there exists an integer n such that bc = np. Now, assume p does not divide b. i.e. there are integers k & i with 0< i < p, such that b = kp + i And therefore, np = bc = c(kp +i ) = kpc + ci ==> ci = p(n - kc) So, p is a factor of ci and since 0 < i < k , p must be a factor of c.
I am particularly confused by the last statement:
... since 0 < i < k, p must be a factor of c
This does not make sense to me... i needs not be less than k.
This proof was selected as the best answer in a Yahoo question: https://answers.yahoo.com/question/index?qid=20121025132152AAzsvU7
I would be grateful if someone could explain me what is going on at the end of the proof or if it is wrong.
Thank you.
