AMC 10 · 2006 · #8
Grade 6 countingPick an answer.
A set of consecutive integers is pinned down by two numbers: where it starts and how many terms it has. Tool #4 (Introduce a Variable) names those as a and n and turns the sum into one clean equation, n(2a+n-1)=30. Tool #14 (Extreme Principle) then caps how long a run can be — the shortest possible run of n positive integers already sums to 1+2+…+n, which cannot exceed 15 — so only a handful of lengths are even possible. Tool #2 (Make a Systematic List) checks those few lengths one by one and counts the winners.
Name the start and the length
Naming the start and the length turns the sum into a product.
A run of consecutive numbers is fixed by where it starts and how many there are, so give both a name.
6.EE.A.2Introduce A VariableCap how long the run can be
Positivity makes the length the smaller factor, so it is at most 5.
The starting number drags the total up fast, so a run that sums to only 15 cannot be very long.
The starting number drags the total up fast, so a run adding to only fifteen cannot be very long.
▸ Why?
Consecutive numbers climb by the same fixed step, so each extra term adds at least as much as the one before.
▸ Why?
Pairing the first term with the last gives the same total as pairing inward, so the sum is the length times the middle.
Test each possible length
Testing each length, one fails to give a whole start.
Only a length that divides 30 evenly and leaves a positive start can actually work.
6.EE.B.7Make A Systematic ListCount the winning sets
Three lengths survive, so the count is 3, choice (C).
Each length gives at most one run, so just tally the ones that survived.
4.OA.B.4Make A Systematic ListDescribe a run of consecutive numbers by where it starts and how many there are, turn the sum into n(2a+n-1)=30, and only a short run can add up to just 15.
- Name the start and the length
- Cap how long the run can be
- Test each possible length
- Count the winning sets