AMC 10 · 2021 · #25
Grade 6 number-theoryPick an answer.
Computing R(n) and R(n+1) and comparing them is the obvious move, and it is the wrong one — it does twice the work and throws away the fact that the two inputs differ by exactly 1. Tool #16 (Change Focus) says to stop tracking the totals and track only the change R(n+1) - R(n). That change splits into nine independent pieces, one per divisor, and tool #5 (Look for a Pattern) settles each piece with a single observation: adding 1 pushes a remainder up by 1, unless it has just hit a multiple, in which case it falls all the way to 0. Tool #13 (Convert to Algebra) then adds the nine pieces into one clean equation whose only ingredient is the list of numbers from 2 to 10 that divide n+1. From there the problem is arithmetic on small sets: tool #2 (Make a Systematic List) writes down every subset of {2,3,…,10} whose sum is 9, tool #3 (Eliminate Possibilities) throws out the ones that no number could actually have as its divisor list, and tool #6 (Guess and Check) screens the handful of survivors in the two-digit range.
Track the change, not the totals
Track the change, not the totals.
Two sums are equal exactly when the difference between them is zero, and a difference of sums can be examined one matching pair at a time.
6.EE.A.2Change Focus Count The ComplementWhat adding one does to a remainder
Adding one changes a remainder in one of two ways.
A remainder counts steps past the last multiple, so it climbs by one each time until it reaches the next multiple and resets to zero.
A remainder counts steps past the last multiple, so it climbs by one each time until it resets to zero.
▸ Why?
Dividing gives one quotient and one remainder smaller than the divisor, and adding one nudges that remainder.
▸ Why?
Once the remainder reaches the divisor it returns to the start, so the pattern repeats on that period.
Add the nine changes together
Add the nine changes together.
Assume the ordinary case everywhere, then pay a penalty exactly at the multiples — the penalty for divisor k is k itself.
6.EE.A.3Convert To AlgebraList every subset summing to nine
List every subset summing to nine.
Sorting the search by how many numbers are used makes it finite fast, because four small numbers already overshoot the target.
4.OA.B.4Make A Systematic ListMost of those sets are impossible
Most of those sets are impossible.
A divisor list is not a free choice of numbers — every divisor brings its own divisors with it, so most lists can never occur.
6.NS.B.4Eliminate PossibilitiesScreen the multiples of fourteen
Screening the multiples leaves 2.
Once the divisor list is pinned down, the survivors are just the multiples of 14 that have stayed clear of every other small factor.
6.NS.B.2Guess And CheckWhen a question compares a quantity at n with the same quantity at n+1, do not compute both — compute only what changes, because adding 1 nudges every remainder up by one except at the multiples, where it crashes to zero.
- Track the change, not the totals
- What adding one does to a remainder
- Add the nine changes together
- List every subset summing to nine
- Most of those sets are impossible
- Screen the multiples of fourteen