AMC 10 · 2025 · #23
Grade 7 number-theoryPick an answer.
The condition names a huge pile of pairwise comparisons, which is hopeless to check one pair at a time. So I first collect all the remainders into a single list and ask what an increasing list can even look like. That is where the whole problem cracks: the list has exactly as many entries as there are available remainders, so it is squeezed with zero slack and is forced to be 0, 1, 2, …, z-1 in order. From there the work turns algebraic — the forced equality becomes a divisibility statement about 2024 — and the leftover job is the routine subproblem of factoring 2024 and adding its divisors.
Collect the remainders into one list
Collect the remainders into one list.
If every neighbouring step goes up, then several steps in a row also go up, so the neighbour comparisons already carry the whole condition.
6.EE.A.2Make A Systematic ListThe list has no room to skip
The slots fit exactly, so each remainder equals x.
Packing z different increasing values into exactly z slots leaves no slack at all, so the list is pinned to 0, 1, 2, and onward.
4.NBT.B.6Extreme PrincipleTurn the forced list into divisibility
That says z divides 2024.
The gap between 2025x and x is just 2024 copies of x, so the whole condition collapses into a question about what divides 2024.
7.EE.A.1Convert To AlgebraCheck the reverse direction
The converse also holds.
If 2024 disappears on division by z, then 2025 behaves like 1, and multiplying by 1 leaves the list in its own natural order.
6.EE.B.5Solve An Easier Related ProblemBreak 2024 into primes
2024 is eight times eleven times 23.
A divisor is nothing more than a choice of how much of each prime to keep, so counting them is a product of a few small choices.
A divisor is nothing more than a choice of how much of each prime to keep, so counting them is a product.
▸ Why?
Every number has exactly one prime recipe, so a divisor is fully described by its exponents.
▸ Why?
Those choices are made independently for each prime, so the numbers of options multiply.
Add the divisors, then drop 1
The divisor sum minus one is 4319.
Expanding the product picks up every divisor exactly once, so one multiplication replaces a sixteen-term addition.
6.EE.A.1Identify SubproblemsWhen an increasing list has exactly as many entries as there are values it can use, it has no room to skip anything and must be 0, 1, 2 and so on in order — and that squeeze turns a scary rule about remainders into the plain question of what divides 2024.
- Collect the remainders into one list
- The list has no room to skip
- Turn the forced list into divisibility
- Check the reverse direction
- Break 2024 into primes
- Add the divisors, then drop 1