Greatest common divisor and least common multiple: building blocks

The intuition behind gcd and lcm, from the perspective of factorizations. For aspiring mathematicians already familiar with prime numbers as the building blocks of integers. Learn the intuition, the definitions, and gain some practice.
This video is appropriate for high school, for an introduction to proof course, for undergraduate mathematics majors, or for the mathematically inclined, especially those interested in cryptography or number theory.

[@godchild7490]

Love this lots.....it really helped me with my math work

[@martinjaskulla5688]

Amazing!

[@jamesedwards6173]

I've liked puzzles my whole life, and this is one of my favorites (encountered long, long ago), which this video reminded me of.
There are two integers, A and B; both are greater than 1. Mathematician S knows their sum. Mathematician P knows their product. (All they know otherwise is what you’ve just been told.) The following conversation ensues.
S says to P: "You don't know A and B."
P says to S: "Now I do know A and B."
S says to P: "Now I know A and B, too."
What are the values of A and B (that result in a minimal sum)?

[@atifachaudhry]

The example at 8:21 of gcd of the two large numbers is stated as 7, yet both these numbers are even, so the gcd should be even.

[@sidharthgopalakrishnan6718]

Regarding the pop quiz question. How is it a yes ? Let me show using a example.
For example if I take 6 and 9 ,
6 = 2 x 3
9 = 3 x 3 hence gcd is 3
but as per your proposition ANY other divisor of a and b ie 2 must also divide gcd ( 6,9) which is false so am I mis understanding your proposition or did you mean ANY common divisor ? Great explanation btw it is really intuitive.

[@marthanelis225]

I think it has to be a common divisor, so 2 should be a divisor of 6 AND 9 to be a divisor of the greatest common divisor. Since 2 isn't a divisor of 9, it indeed does not divide the gcd (6,9)