Number Theory

https://www.youtube.com/watch?v=8Dlkur8Ea-k Euclid’s lemma is a famous result in number theory which states that, if a prime divides a product of integers, then the prime has to divide at least one of those integers. In this video, we prove a generalization of this fact called Gauss’s divisibility lemma, and deduce Euclid’s lemma from it. The proof […]

https://www.youtube.com/watch?v=PrswtmYucAk It is well-known that there are infinitely many prime numbers, but how can we prove that this is true? In ancient Greece, Euclid had a proof which is still the best one. First we start with the lemma that every integer greater than 1 has a prime factors. Then we use this to prove

https://www.youtube.com/watch?v=gYE_G8z-5uQ Given a tuple or list of integers, there are intimate connections between their greatest common divisor or GCD, their collection of common divisors, and their linear combinations. In particular, the smallest positive linear combination is in fact the GCD, the common divisors are the divisors of the GCD, and the linear combinations are the

https://www.youtube.com/watch?v=6P-h2OWE-tk Most of us have done long division in school, where we divided one number by another to find the quotient and remainder. Why do the quotient and remainder always exist uniquely? The answer is given by Euclidean division, otherwise known as the division algorithm. We prove the Euclidean division theorem here by showing the