m = 4n + 9, where n is a positive integer. What is the greatest common factor of m and n?

(1) m=9s, where s is a positive integer

(2) n?=4t, where t is a positive integer

In the solution discussing point (1): if (1) is true, then n itself must be a multiple of 9. I have understood that.

What I did not understand is the next sentence: "If both m and n are multiples of 9, and m is exactly 9 units away from 4n, then the largest possible common factor is 9". Can you prove that besides listing cases?

I can't get toward a proof or I can get the intuition behind this.