Competition · AMC preparation · step 4 of 4
AMC 8 · 2020 · #12
Grade 4 algebranumber-theoryPick an answer.
AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #7 (Identify Subproblems) splits the equation 5! · 9! = 12 · N! into three small, friendly pieces: (a) compute 5! since it is the only "small" factorial in sight, (b) divide both sides by 12 to clear the loose coefficient, and (c) read off the surviving expression as a single factorial. Each subproblem is a Grade 3-4 arithmetic move; the big-looking factorial expression never has to be expanded. Tool #3 (Eliminate Possibilities) is a natural backup for any AMC multiple-choice question — once we have 10 · 9! = N!, the five choices 10, 11, 12, 13, 14 can each be tested against the recursive rule n! = n · (n-1)! until only one fits.
Evaluate 5 factorial
Only 5! is small enough to expand — 9! we leave untouched — and its value is 120.
Multiplying five one-digit numbers together is exactly the Grade 4 multi-digit multiplication skill — no factorial magic needed.
4.NBT.B.5Identify SubproblemsSubstitute back into the equation
Substitute 5! = 120 into the equation to get 120 · 9! = 12 · N!.
Treating 9! and N! as "mystery numbers" turns the problem into a familiar Grade 3 "unknown factor" equation.
3.OA.A.4Identify SubproblemsDivide both sides by 12
Divide both sides by 12 (since 120 ÷ 12 = 10) to get 10 · 9! = N!.
Dividing 120 by 12 to get 10 is a basic Grade 3 multiplication/division fact within 100.
3.OA.C.7Identify SubproblemsRewrite the product as a factorial
By the rule n! = n · (n-1)!, the left side folds into one factorial: 10 · 9! = 10!, so 10! = N!.
Spotting the rule "next factorial = next number × previous factorial" is the Grade 4 "generate a number pattern following a given rule" standard.
The whole left-hand side collapses into a single factorial: 10 · 9! = 10!.
▸ Why?
By the given meaning of factorial, 9! = 9 × 8 × … × 1, so 10 · 9! is 10 × (9 × 8 × … × 1), i.e. the one unbroken product 10 × 9 × 8 × … × 1 — and that is exactly what 10! means: every integer from 10 down to 1 multiplied together.
▸ Why?
Erasing the parentheses around 9 × 8 × … × 1 and reading 10 × 9 × 8 × … × 1 as one long chain is allowed, because in a string of multiplications you may group any two factors first without changing the product.
Match the two factorials
Equal factorials of positive integers force equal integers, so N = 10 — choice (A).
"Same product, same factor list, so same top number" is the Grade 3 "find the unknown in a multiplication equation" idea, used here to eliminate the other four choices.
3.OA.A.4Eliminate PossibilitiesThis AMC 8 problem only needs Grade 4 multiplication patterns — like 10 × 9! = 10! — that you already know!
- Evaluate 5 factorial
- Substitute back into the equation
- Divide both sides by 12
- Rewrite the product as a factorial
- Match the two factorials
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