AMC 10 · 2019 · #4
Grade 6 algebraPick an answer.
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).
Factor out the common piece
Both terms share the same factorial.
Factoring out the shared n! turns a factorial puzzle into a small algebra puzzle.
Factoring out the shared factorial turns a factorial puzzle into a small algebra puzzle.
▸ Why?
A factor shared by every term can be lifted out front, leaving something small inside.
▸ Why?
A factorial is the next one with a couple of extra factors stacked on, so the shared piece is easy to spot.
Tidy the leftover
It tidies into a product of two numbers.
Two whole numbers that differ by 2 multiply to 440 — much easier to chase.
6.EE.A.3Solve An Easier Related ProblemSplit 440 into two factors
The two factors differ by two.
√(440)≈ 21, so the two factors hug 21 — try 20 and 22.
4.OA.B.4Guess And CheckSolve for n
That gives n immediately.
Both equations agree, so n = 19 is the right fit.
6.EE.B.7Guess And CheckAdd the digits
Adding the digits gives 10.
Digit sum of a two-digit number is just tens-place plus ones-place.
2.NBT.A.1Eliminate PossibilitiesThis AMC 12 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.