AMC 10 Number Theory: Algebra to Diophantine Trick

Watch how a single AMC 10 counting problem collapses into a clean Diophantine equation once you square both sides and factor 49. Dr. Ashani Dasgupta walks through the full difference-of-squares route — and shows exactly why there are precisely six integer solutions, not five and not seven.

What you'll learn:
✦ How to clear a radical and rearrange algebra into a solvable Diophantine form
✦ The difference-of-squares move that turns the equation into a product equal to 49
✦ Why listing the integer factor pairs of 49 hands you every solution systematically
✦ How to recover each ordered pair by solving the resulting linear system

⏱ Chapters
00:00 — Why algebra-to-Diophantine matters
00:47 — Problem setup: counting integer ordered pairs
01:32 — Square both sides and clear the radical
02:46 — The difference-of-squares factorization
03:28 — Why the product must equal 49: listing factor pairs
04:13 — Solving each system for the ordered pairs
05:06 — Recommended reading: Mathematical Circles (Fomin)

About Cheenta:
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Видео AMC 10 Number Theory: Algebra to Diophantine Trick канала Cheenta Academy for Olympiad & Research