Can You Solve This "Impossible" Math Olympiad Problem?

This problem seems impossible: prove a number has at least 4^n divisors without even knowing its prime factors! This is a classic Math Olympiad-style challenge that tests deep number theory concepts. Watch as we reveal the beautiful secret of cyclotomic polynomials and Zsigmondy's Theorem to solve it step-by-step. Stick around for the mind-blowing bonus content where we show why the conditions of the problem are absolutely essential. Can you solve it before we do? Timestamps: 00:00 - The "Impossible" Olympiad Problem 00:33 - The Core Strategy: Building Divisors 01:38 - The Subset Trick: Finding a Family of Divisors 03:23 - The Key Insight: Decomposing with the GCD Lemma 05:25 - The Final Count & Zsigmondy's Theorem 08:32 - Sanity Check: Does the Math Actually Work? 09:25 - BONUS: The Alternative "Prime Counting" Method 10:17 - FINAL BONUS: Why p greater than 3 is Not a Suggestion, It's the Law