In this video we cover the following concepts:

gcd - We prove that \(\gcd(ab,c) = \gcd(a,c)\gcd(b,c)\) if \(\gcd(a,b,c) = 1\).

more gcds - We use the above formula to prove that if \(\gcd(m,n) = 1\) and if \(D \mid mn\) then \(D = \gcd(m,D) \gcd(n,D)\) so that \(D\) factors as a divisor of \(m\) times a divisor of \(n\).

Let’s make a table with \(m = 20\) and \(n = 21\), so \(mn = 420\)

\(d\) \(\gcd(d,20)\) \(\gcd(d,21)\)
1 1 1
2 2 1
3 1 3
4 4 1
5 5 1
6 2 3
7 1 7
10 10 1
12 4 3
14 2 7
15 5 3
20 20 1
21 1 21
28 4 7
30 10 3
35 5 7
42 2 21
60 20 3
70 10 7
105 5 21
140 20 7
210 10 21
420 20 21

Now ask yourself, what is the relationship between the number of divisors of \(20\) (which is \(6\)) and the number of divisors of \(21\) (which is \(4\)) and the number of rows of the above table.