AMC 10 · 2002 · #13
Grade 8 number-theoryPick an answer.
Every run of 18 consecutive integers is decided by one number: where it starts. Tool #4 (Introduce a Variable) names that starting number a and turns the whole family of runs into a single formula for the total. Tool #15 (Organize Information in More Ways) then rewrites that formula in factored form, which is where the perfect-square condition becomes readable. Tool #9 (Solve an Easier Related Problem) shrinks the question "is this total a square?" to a much smaller question about one factor. Tool #14 (Extreme Principle) picks the smallest legal value of that factor, and Tool #6 (Guess and Check) confirms the winner by adding the actual run.
Name the starting number
With a as the smallest, the eighteen terms total 18a + 153.
One letter for the start captures every possible run at once, so you study a formula instead of infinitely many lists.
6.EE.A.2Introduce A VariableFactor out the nine
Both parts are multiples of nine, so the total is 9(2a+17).
Rewriting a sum as a product is what lets you see squares, because squares are built out of repeated factors.
7.EE.A.2Organize Information In More WaysShrink the square condition
Since nine is already a square, the total is a square exactly when 2a+17 is a square.
A square factor you already have can be peeled off for free, leaving a smaller question of exactly the same kind.
A square factor already present can be peeled off, leaving a smaller question of exactly the same kind.
▸ Why?
Every number breaks into primes in one way, so a square is exactly a number whose primes all appear an even number of times.
▸ Why?
The nine came out of the sum as a shared factor, which is what exposed the square in the first place.
Take the smallest legal square
That value is odd and at least 19, so the smallest legal square is 25.
When the total marches up in step with one factor, minimizing the total means minimizing that factor.
7.EE.B.4Extreme PrincipleCheck the winning run
Then a = 4 and the run 4 through 21 totals 225, choice (B).
Producing the actual run turns a lower bound into a fact — the value is not just allowed, it happens.
4.OA.A.3Guess And CheckEighteen numbers in a row always add to 9 times an odd number, so the sum is a square exactly when that odd number is one.
- Name the starting number
- Factor out the nine
- Shrink the square condition
- Take the smallest legal square
- Check the winning run