Competition · AMC preparation · step 4 of 4
AMC 10 · 2019B · #6
Grade 6 algebraPick an answer.
AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Factorials are scary, but the same n! hides in every term. Divide it out (Tool #9 simplifies the equation), look at the shape that remains (Tool #5: it becomes a product of two consecutive-step integers), then try a few values (Tool #6) and confirm by checking the answer choices' digit sums (Tool #3).
Pull out the common factorial
Every term hides the same n!, so factor that common n! out of the whole equation.
Factoring out the shared n! turns a factorial puzzle into a small algebra puzzle.
Pulling out the shared factorial turns a factorial puzzle into a small algebra puzzle.
▸ Why?
A factor shared by every term can be lifted out of the whole expression at once.
▸ Why?
A factorial is the next one down with extra factors stacked on, so the shared piece is easy to spot.
Cancel n factorial
Cancel n! from both sides; what remains factors to (n+1)(n+3) = 440.
Two whole numbers that differ by 2 multiply to 440 — much easier to chase.
6.EE.A.3Solve An Easier Related ProblemFind two factors of 440
Two factors two apart multiply to 440; near √440 ≈ 21, the pair 20 × 22 works.
√(440)≈ 21, so the two factors hug 21 — try 20 and 22.
4.OA.B.4Guess And CheckMatch the pair to n
Set n+1 = 20 (and n+3 = 22 agrees), giving n = 19.
Both equations agree, so n = 19 is the right fit.
6.EE.B.7Guess And CheckAdd the digits
The digits of n = 19 add to 1 + 9 = 10, which is choice (C).
Digit sum of a two-digit number is just tens-place plus ones-place.
2.NBT.A.1Eliminate PossibilitiesThis AMC 10 problem only needs Grade 6 factoring you already know: pull the shared n! out of both terms, get (n+1)(n+3)=440, spot 20 × 22, so n=19 and the digit sum is 10.
- Pull out the common factorial
- Cancel n factorial
- Find two factors of 440
- Match the pair to n
- Add the digits
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