AMC 10 · 2002 · #17
Grade 5 number-theoryPick an answer.
"Smallest possible" is exactly the signal for Tool #14 (Extreme Principle), and a minimum question always splits into two halves via Tool #7 (Identify Subproblems): prove a floor that no set can go under, then exhibit one set that sits exactly on the floor. Tool #15 (Organize Information in More Ways) supplies the floor — stop reading the set as a list of numbers and read it as nine digits, each charged according to its place, so the question becomes which digits are forbidden from ending a prime. Tool #6 (Guess and Check) closes the argument by producing an actual set of primes that pays exactly that price; without it, a floor is only a bound, not an answer.
Write the sum by place value
The total is each digit weighted by its place value, so push every digit as far right as possible.
Sliding a digit one place to the left multiplies what it adds to the total by ten, so a cheap total keeps every digit as far right as the rules permit.
A digit costs ten times as much one place to the left, so a small total keeps every digit as far right as the rules allow.
▸ Why?
Each place bundles ten of the place to its right, so moving a digit left multiplies what it contributes by ten.
▸ Why?
The sum is what every digit's place amount comes to together, so each digit can be priced on its own.
Three digits cannot end a prime
A prime never ends in 4, 6 or 8, and none is prime alone, so those three cost at least the tens place.
An even last digit makes the whole number even, and the only even prime is 2 — so these three digits get pushed left no matter what.
4.OA.B.4Extreme PrincipleAdd up the floor: 207
Charging the minimum everywhere gives 180 + 27 = 207 as a floor no set can beat.
Charging every digit the cheapest position it is legally allowed gives a total that nothing can slip under.
4.NBT.B.4Extreme PrincipleLand on the floor with an example
The set {41, 67, 89, 2, 3, 5} is all primes, uses each digit once, and sums to 207, choice (B).
A lower bound turns into the true minimum the instant you exhibit one set that matches it exactly.
4.OA.B.4Guess And CheckTo find a smallest, first prove a floor that nothing can go under, then build one example that lands exactly on it.
- Write the sum by place value
- Three digits cannot end a prime
- Add up the floor: 207
- Land on the floor with an example
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