AMC 10 · 2020 · #6

Grade 6 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
factorialpolynomial-factoringperfect-squares identify-subproblemsconvert-to-algebraguess-and-check ↑ Prerequisites: factorialpolynomial-factoring
📏 Medium solution 💡 2 insights
Problem
Take any integer at least 9. Build the expression that subtracts the factorial of the number plus one from the factorial of the number plus two, then divides by the factorial of the number itself. Four of the five listed descriptions fail for some value; find the one that fits for every allowed integer.

Pick an answer.

(A)
a multiple of 4
(B)
a multiple of 10
(C)
a prime number
(D)
a perfect square
(E)
a perfect cube
How to solve
Strategy Organize Information in More Ways

The three factorials n!, (n+1)!, and (n+2)! look like three unrelated giant numbers, but they are really the same product written to three different lengths. Rewriting all of them in terms of the shortest one, n!, is the move that makes the whole fraction collapse — that is why the main tool is re-organizing the given information rather than computing. After the collapse, factoring the leftover expression is a small subproblem on its own. Finally, since the question asks what is always true, one well-chosen test value plus the closed form settles which description survives and which four die.

1STEP 1

Write every factorial one way

Write them all over the same factorial.

(n+1)! = (n+1) · n!, (n+2)! = (n+2)(n+1) · n!
2STEP 2

Cancel the shared factorial

The shared factorial cancels entirely.

((n+2)!-(n+1)!)/n! = ((n+2)(n+1) n! - (n+1) n!)/n! = (n+2)(n+1) - (n+1)
3STEP 3

Factor what remains

What remains is a perfect square.

(n+2)(n+1) - (n+1) = (n+1)[(n+2)-1] = (n+1)(n+1) = (n+1)²
4STEP 4

Test one value

Check with one value.

n = 9: (11!-10!)/9! = 36288000/362880 = 100 = 10²
5STEP 5

Pick the surviving description

The answer is that it is a perfect square.

(n+1)² is a perfect square for every integer n; n = 10 → 121
Answer
a perfect square
The closed form (n+1)² passes every sanity test. It is positive and grows fast, which matches a difference of factorials divided by a smaller factorial. It agrees with the direct computation at n = 9, where both give 100, and a second check at n = 10 gives (12!-11!)/10! = (479001600-39916800)/3628800 = 439084800/3628800 = 121 = 11², exactly as predicted. The value 121 also shows why no other choice can be right: it is odd, ends in 1, factors as 11 · 11, and sits strictly between 4³ and 5³, so it breaks (A), (B), (C), and (E) simultaneously.
💡Key takeaway

Rewrite every factorial in terms of the smallest one, cancel it, and the monster fraction shrinks to (n+1)² — always a perfect square.

  • Write every factorial using n!
  • Cancel the shared n!
  • Factor out the common (n+1)
  • Test one value to confirm
  • Match the survivor to a choice

More like this

Same archetype — closest grade level first.