AMC 10 · 2023 · #22
Grade 9 number-theoryPick an answer.
There is no formula for f to plug 2023 into, so the only handle is the family of equations itself. Two observations turn that family into something finite. First, rewriting the sum so the d=1 term sits alone (Organize Information in More Ways) shows that the equation for n contains f(n) exactly once, with everything else evaluated at strictly smaller divisors of n. That single rearrangement is why f exists and is unique, and it turns the condition into a recipe that computes f(n) from values already known. Second, the equation holds for every n, so nothing stops you from using tiny values of n first (Solve an Easier Related Problem) and climbing upward. Since only divisors of 2023 ever appear along the way, and 2023 = 7 · 17² has just six divisors, the climb is six short steps (Identify Subproblems). Along the way a pattern shows up for prime powers (Look for a Pattern), which is worth noticing but is not needed — the six-step climb reaches the answer on its own, and it avoids the multiplicativity shortcut that quietly produces one of the wrong choices (Eliminate Possibilities).
Isolate the term
Isolate the term you want.
Every term except the d=1 one looks at a smaller number, so the equation for n has exactly one genuinely new unknown in it.
9.F-IF.A.2Organize Information In More WaysList every divisor
There are only six divisors.
A divisor of a divisor is still a divisor, so the whole computation stays trapped inside the six divisors of 2023.
4.OA.B.4Identify SubproblemsStart at the bottom
Start from the smallest.
The number 1 has only itself as a divisor, so its equation has just one term and hands over the starting value for free.
9.F-IF.A.3Solve An Easier Related ProblemThe values at the primes
Compute the values at both primes.
A prime has only two divisors, so its equation is f(p) plus p times the known f(1), and it unwinds in one move.
6.EE.B.5Solve An Easier Related ProblemAt the square of a prime
Compute it at the prime's square.
The two known terms -272 and +289 almost wipe each other out, leaving the same value the prime itself had.
7.NS.A.2Look For A PatternAt the product of two primes
Compute it at their product.
The divisor d and the input n/d trade places, so the small divisor multiplies the value at the big one.
A divisor and the quotient it leaves behind trade places, so the small divisor pairs with the big one.
▸ Why?
Divisors come in pairs that multiply back to the number, so naming one names the other.
▸ Why?
Every number has exactly one prime recipe, so its divisors form a short and complete list.
Finish at the target
The final value is 96.
Six divisors means six terms, five of them already known, so the last unknown is whatever makes the total land on 1.
6.EE.A.2Identify SubproblemsWhen a rule holds for every number, try the smallest numbers first — each equation hands you one new value, and you climb from there.
- Isolate the term holding f(n)
- Only six numbers ever matter
- Start at the bottom: n=1
- Climb to the primes 7 and 17
- Square of a prime: f(289)
- Product of two primes: f(119)
- Finish at n = 2023