AMC 10 · 2023 · #24
Grade 8 number-theoryPick an answer.
Seven multiplicative conditions tangle four unknown numbers together, and there is no useful place to start while they stay whole numbers. Tool #7 (Identify Subproblems) is the whole move: switch from the numbers to their prime exponents, and the seven conditions fall apart into three puzzles that never mention each other, one for the twos, one for the threes, one for the fives. Tool #4 (Introduce a Variable) makes that switch possible by naming the twelve exponents. Tool #15 (Organize Information in More Ways) supplies the dictionary that does the translating: multiplying adds exponents, an lcm takes the larger exponent, a gcd takes the smaller. Inside each small puzzle, tool #14 (Extreme Principle) starts at the tightest condition, because the lowest ceiling is the one that pins other numbers down. Tool #3 (Eliminate Possibilities) then removes a capped number from contention whenever a high maximum has to be reached by somebody. Tool #2 (Make a Systematic List) closes each prime by sweeping the handful of exponent quadruples that survive, which also proves the answer does not depend on which one is chosen.
Turn them into exponents
Turn each into three exponents.
A number built only from 2s, 3s and 5s is completely described by how many of each it holds.
4.OA.B.4Introduce A VariableTranslate the operations
Products add; the lcm takes the larger.
lcm and gcd look complicated on numbers, but on exponents they are only "take the bigger" and "take the smaller".
On exponents a least common multiple is only take the bigger and a greatest common divisor only take the smaller.
▸ Why?
A common multiple must contain every prime at least as often as each number does, and a common divisor at most.
▸ Why?
Each number has exactly one prime recipe, so the primes never interfere with one another.
Three primes, three puzzles
Each prime is a separate puzzle.
Nothing in the problem ties the twos to the threes, so they never have to be untangled together.
6.EE.A.1Identify SubproblemsSolve the first prime
Start from the lowest ceiling.
The lowest ceiling is the most informative: once a number is capped low, every higher maximum in its rows has to be reached by its partner.
6.EE.B.8Extreme PrincipleSolve the second prime
A low cap forces the values.
Ruling two numbers out of a high maximum leaves the high value with nowhere to sit but the other two.
6.EE.B.5Eliminate PossibilitiesSolve the third prime
The third works the same way.
Three pairwise maxima all equal to 2 cannot be carried by a single number, because every one of the three pairs needs a 2 inside it.
7.EE.B.4Make A Systematic ListRebuild the answer prime by prime
Rebuilding gives 3.
The three subproblems were solved apart, so the answer has to be glued back together one prime at a time.
6.NS.B.4Identify SubproblemsWhen a problem is built out of multiplying, lcm and gcd, stop looking at the numbers and count prime factors instead: each prime becomes its own small puzzle, and lcm just means take the bigger count.
- Turn each number into three exponents
- Multiplying adds, lcm takes the larger
- Three primes, three separate puzzles
- Prime 2: the tightest ceiling first
- Prime 3: one low cap forces two threes
- Prime 5: two of the three must be 2
- Rebuild the gcd prime by prime