AMC 10 · 2009 · #13
Grade 8 algebraPick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Everything is written with the two building blocks P = 2^m and Q = 3ⁿ. Break the base 12 into its prime powers of 2 and 3, then rebuild each prime power out of P and Q using the exponent rules. A quick number test at the end pins down the single matching choice.
Factor the base into primes
The base 12 is 2² × 3, and a power of a product powers each factor, so the twos and the threes separate.
Splitting 12 into its 2-part and 3-part lets us handle the twos and threes separately.
8.EE.A.1Introduce A VariableRebuild the prime powers from P and Q
A power of a power multiplies exponents: P²ⁿ = 2²mn and Q^m = 3^mn — exactly the two pieces needed.
Stacking one exponent on another multiplies them, so P and Q can reach any exponent we want.
Stacking one exponent on another multiplies them, so the two letters can reach any exponent needed.
▸ Why?
An exponent counts how many times a factor is used, so using a block several times multiplies those counts.
▸ Why?
Every number has exactly one prime recipe, so the twos and threes can be handled quite separately.
Match the pieces together
The split 2²mn · 3^mn is just P²ⁿ times Q^m, so 12^mn is rebuilt from P and Q alone.
Once each prime power is written with P or Q, the target is just their product.
6.EE.A.2Organize Information In More WaysConfirm with a quick number test
Test m=2, n=1: the target is 144, matched by 4²·3² but not by the look-alike P²Q = 4²·3 = 48.
Plugging in small numbers confirms the algebra and knocks out look-alike choices.
6.EE.A.1Eliminate PossibilitiesBreak a number into its prime powers, then rebuild the target one prime at a time using the exponent rules.
- Factor the base into primes
- Rebuild the prime powers from P and Q
- Match the pieces together
- Confirm with a quick number test