AMC 10 · 2013 · #9
Grade 6 number-theoryPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #7 (Identify Subproblems) splits the task into three clean jobs: break 27000 into prime powers, decide how those primes get grouped into three numbers, then add. The middle job is where Tool #3 (Eliminate Possibilities) does the heavy lifting: the pairwise-coprime rule forbids two numbers from sharing any prime, so a prime's full power cannot be split across two factors. That single rule eliminates every grouping except one, so no guessing is needed.
Prime factorize 27000
Since 27000 = 27 × 1000, the prime factorization is 2³ × 3³ × 5³ — only the primes 2, 3, and 5 appear.
Every whole number breaks into primes in exactly one way, so factoring reveals all the pieces you have to work with.
4.OA.B.4Identify SubproblemsCoprime rule forces the grouping
Splitting 2³ would leave two even numbers sharing the factor 2, so each prime block stays whole: one number each.
If a prime were spread across two numbers, they would share that prime, so each prime must stay whole in one number.
If a prime were spread across two numbers, they would share that prime, so each prime stays whole in one.
▸ Why?
Every number has exactly one prime recipe, so a shared prime shows up in both factor lists.
▸ Why?
That shared prime would then divide both numbers, breaking the no-common-factor rule.
Compute the three numbers
Evaluating the blocks gives 8, 27, and 125, all above 1, and 8 × 27 × 125 = 27000 confirms the product.
A power like 5³ just means multiplying the base by itself three times.
6.EE.A.1Identify SubproblemsAdd the three numbers
Adding them gives 8 + 27 + 125 = 160, which only choice (D) matches, so the rest are eliminated.
Once the three numbers are pinned down, the sum is a single addition.
4.NBT.B.4Eliminate PossibilitiesWhen numbers must be pairwise relatively prime, each prime's full power has to stay inside one number — so factor into primes, hand out one prime block per number, then add.
- Prime factorize 27000
- Coprime rule forces the grouping
- Compute the three numbers
- Add the three numbers