AMC 10 · 2024 · #11
Grade 8 algebranumber-theoryPick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Squaring the equation turns √(n² - 49) = m into n² - m² = 49. Tool #2 (Make an Organized List) is the natural way to hunt for integer solutions: walk through small candidate values of n with |n| ≥ 7 and check whether n² - 49 is a perfect square. Tool #6 (Guess and Check) is what each test inside the list actually does. Tool #16 (Count the Complement) helps us avoid double work: because the equation only mentions n² and m², every solution with positive n has a twin with negative n, so we can count positive n first and double when appropriate.
Square both sides to clear the radical: since √ returns a non-negative value equal to m, we have m ≥ 0, so squaring is safe.
Grade 8 treats the square root symbol as the inverse of squaring a non-negative number — squaring both sides is the standard move to clear it.
8.EE.A.2Guess And CheckReframe as a search: find integers n making n² - 49 a perfect square; the radical forces |n| ≥ 7, so start the list at n = 7.
Grade 6 framing of "solve an equation" as testing values that make both sides equal — here we test which n make n² - 49 land on a perfect square.
6.EE.B.5Make A Systematic ListTest n = 7, 8, 9, … : once the gap 2n - 1 ＞ 49 (n ≥ 26) no square can land, so only n = 7 and n = 25 work.
Grade 4 recognizing perfect squares (0, 1, 4, 9, …, 576) is enough to scan the list. The gap argument prunes the search to a finite range.
4.OA.B.4Make A Systematic ListMirror to negative n: only n² appears, so n and -n pair up — the two positive-n solutions double into four ordered pairs.
Grade 6 idea that a number and its opposite have the same square — so every positive-n solution is paired with a negative-n twin.
6.NS.C.6Count The ComplementWhen a square root has to equal an integer, square both sides and hunt: which integers make the inside a perfect square? Bounding the search with the gap between consecutive squares turns an infinite-looking problem into a quick finite list.