Olympiad Number Theory for Beginners - HOW to Solve IMO 2023 Problem 1

The 2023 International Mathematical Olympiad took place in Chiba, Japan. This video is about Problem 1 - a number theory problem. Problem 1 is generally the most accessible out of the 6, and the same was true this year. But it’s still an IMO problem, so unless you’re a well-trained competitor, it’s no walk in the park!
For a written solution, see this article:

[justsomeboyprobablydressed]

I don't understand your proof. It doesn't even rule out 12. At 7:19, you said, if d_2, and d_3 are distinct primes, then n/d_2 is not a multiple of d_2. But n = 12 is a counterexample to that. The factors are 1, 2, 3, 4, 6, 12. We have d_2 = 2, and so n/d_2 = 6. But 6 is a multiple of 2. So, n/d_2 is a multiple of d_2.