AMC 10 · 2016 · #16
Grade 6 countingPick an answer.
Name the length of the sequence and its starting number. That turns the sum into one clean equation. The equation factors into a product equal to a fixed number, so counting solutions becomes counting factor pairs of that number — a finite list we can check.
Name the start and the length
Two letters describe every run.
Once you name the first term and how many terms there are, the whole list is pinned down.
6.EE.B.6Introduce A VariableWrite the sum as one equation
The sum becomes one clean product.
A run of consecutive numbers equals its middle value times how many there are, which collapses the sum into a single product.
A run of consecutive numbers equals its middle value times how many there are.
▸ Why?
Pairing the first with the last gives the same total as pairing inward, so every pair matches.
▸ Why?
That common pair total is twice the middle value, which is what an average of the run means.
Read off the two factors
That number factors into four primes.
Splitting 690 into k times something turns the whole problem into choosing a factor.
4.OA.B.4Identify SubproblemsList the factor pairs of 690
There are eight factor pairs in all.
Each way to split 690 into two factors is one candidate sequence.
4.OA.B.4Make A Systematic ListDrop the length-1 case and count
Dropping the single-term case gives 7, choice (D).
Every allowed split gives exactly one real sequence, so counting the splits counts the answers.
6.EE.B.7Make A Systematic ListA run of consecutive numbers equals its middle value times its length, so turning the sum into a product lets you just count factor pairs.
- Name the start and the length
- Write the sum as one equation
- Read off the two factors
- List the factor pairs of 690
- Drop the length-1 case and count