AMC 10 · 2005 · #17
Grade 11 algebranumber-theoryPick an answer.
A sum of logarithms is hard to reason about directly, so we convert it into algebra about exponents (Tool #13): exponentiating turns the equation into 2^a3^b5^c7^d = 10²⁰⁰⁵. Rational exponents are still awkward — unique factorisation only compares integer exponents — so we introduce a common denominator N (Tool #4) and raise both sides to the N-th power, reducing the question to an easier related problem (Tool #9) about integers only. Then reorganising both sides into prime factorisations (Tool #15) forces the comparison. The step that must not be skipped is the one that looks obvious: "there is no 3 on the right, so b=0" is the whole claim, not a free move, and it is only legitimate once the exponents are integers.
Move coefficients into exponents
Each coefficient moves into an exponent.
A logarithm turns multiplying into adding, so a coefficient in front of a log is an exponent in disguise.
11.F-LE.A.4Convert To AlgebraCollapse to one power equation
The sum collapses into one power equation.
Equal logarithms mean equal numbers, so the problem becomes a statement about a product of prime powers.
11.F-LE.A.4Organize Information In More WaysClear the denominators
A common denominator makes every exponent an integer.
Rational exponents follow the same rules as integer ones, so a single common denominator turns the whole equation into a whole-number statement.
11.N-RN.A.1Introduce A VariablePush everything to one side
Moving everything to one side leaves a product equal to one.
Comparing against 1 is easier than comparing two large products — now every prime has to cancel itself out.
8.EE.A.1Solve An Easier Related ProblemMatch the prime factorisations
Disjoint primes force every exponent to be zero.
Two numbers built from completely different primes can only be equal if both are 1.
Two numbers built from completely different primes can only be equal when both of them are 1.
▸ Why?
Every number has exactly one prime recipe, so equal numbers must use the same primes with the same counts.
▸ Why?
An exponent records how many times a prime is used, so matching the recipes matches the exponents one by one.
Read off the tuple
So the whole tuple is forced, with no freedom anywhere.
Once the integer exponents are pinned down, dividing by the common denominator N recovers the original rational numbers.
9.A-REI.B.3Introduce A VariableCheck it, then count
It checks out, so exactly 1 tuple exists, choice (B).
Existence and uniqueness are two separate jobs: substitution gives one, prime factorisation gives the other.
11.F-LE.A.4Convert To AlgebraOne equation with four unknowns usually has endless solutions, but primes are stubborn: 2, 3, 5 and 7 cannot imitate each other, so only a single tuple survives.
- Move coefficients into exponents
- Collapse to one power equation
- Clear the denominators
- Push everything to one side
- Match the prime factorisations
- Read off the tuple
- Check it, then count