Competition · AMC preparation · step 4 of 4
AMC 10 · 2024A · #15
Grade 8 number-theoryalgebraPick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The phrase "both ... are perfect squares" names two unknown squares with a fixed difference — exactly the trigger for Tool #13 (Convert to Algebra): give the two squares names a² and b², subtract to eliminate M, and the unknowns are now an integer pair (a,b) with b² - a² = 2560. The difference of squares factors that into (b-a)(b+a) = 2560, turning the squares problem into a factor-pair problem. Tool #7 (Identify Subproblems) then splits the work cleanly: first find a from the right factor pair (max M comes from the most lopsided pair), then turn a into the units digit using only the ones-digit arithmetic of a² - 1213.
Subtract to remove M
Name the two squares M + 1213 = a² and M + 3773 = b², subtract to cancel M, leaving a fixed gap of 2560.
Writing both "perfect square" conditions as equations lets you cancel the unknown M — the kind of move Grade 6 introduces when it teaches expressions with variables.
6.EE.A.2Convert To AlgebraFactor the difference of squares
Factor it as (b-a)(b+a) = 2560; same parity plus an even product means both must be even.
Difference of squares is the algebra version of arranging b² - a² as an L-shaped strip whose two side lengths are b-a and b+a. Same-parity is the integer-factorization constraint that comes for free.
The difference of two squares factors into their difference times their sum.
▸ Why?
The two cross terms are opposites, so they wipe each other out and only the squares remain.
▸ Why?
Only factor pairs of the resulting number can work, so the candidates form a short closed list.
Pick the best factor pair
Pushing the even factors farthest apart gives 2 × 1280, so b = 641 and a = 639.
For a fixed product, two numbers are pulled apart most when one of them is as small as the rules allow. Even-only forces the smaller factor down to 2, not 1.
4.OA.B.4Identify SubproblemsFind the units digit
Units digits only need units digits: 9² ends in 1, minus the 3 from 1213 gives units digit 8.
Units digits only depend on units digits — that is the whole reason mod-10 arithmetic exists. No need to compute the full 639² = 408,321.
8.EE.A.1Identify SubproblemsWhen two perfect squares have a fixed difference, factor that difference — the factor pair you pick is exactly the system b-a = x, b+a = y that solves for the two square roots. Lopsided pairs give the largest squares.
- Subtract to remove M
- Factor the difference of squares
- Pick the best factor pair
- Find the units digit
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