AMC 10 · 2003 · #14
Grade 8 number-theoryPick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The number is already in prime-factored form 3⁸ · 5², so writing it as a^b means pulling the exponent b outside a base a. Tool #4 (Introduce a Variable) pins down what b is allowed to be: b must divide the exponent of every prime, so it can only be a common divisor of 8 and 2. That leaves just two choices for b, and tool #3 (Eliminate Possibilities) checks each. Because a bigger exponent b makes the base a far smaller — and a is what dominates the sum — tool #14 (Extreme Principle) says the smallest a + b comes from pushing b to its largest allowed value. That single idea points straight at the answer.
See what "a to the b" demands
Writing 3⁸ · 5² = a^b splits each prime's exponent into b equal groups, so b must divide both 8 and 2.
A power a^b hands each prime's exponent out equally to b copies, so b has to divide every exponent.
A power hands each prime's exponent out equally to every copy, so the exponent has to divide them all.
▸ Why?
An exponent counts how many times a factor is used, so raising to a power multiplies every count.
▸ Why?
Every number has exactly one prime recipe, so the exponents cannot be shuffled between primes.
List the only exponents allowed
Since gcd(8, 2) = 2, the common divisors of 8 and 2 are just 1 and 2, so b is 1 or 2 — only two cases.
An exponent that must divide both 8 and 2 can be no larger than their greatest common factor.
6.NS.B.4Eliminate PossibilitiesTake the largest exponent, b = 2
Halving each exponent gives the base a = 3⁴ · 5 = 405, and 405² = 3⁸ · 5² confirms it works.
Halving every exponent turns the number into a perfect square whose root is 3⁴ · 5 = 405.
8.EE.A.1Extreme PrincipleCompare the two cases and pick the smaller sum
b = 2 gives 405 + 2 = 407, while b = 1 gives 164025 + 1 = 164026, so the minimum is 407, choice (D).
A bigger exponent pulls the base down sharply, and the base drives the sum, so the biggest allowed b gives the smallest total.
6.EE.A.1Eliminate PossibilitiesTo pack a number into a^b with the smallest a + b, use the biggest exponent that divides every prime's power — a larger exponent shrinks the base far more than it adds.
- See what "a to the b" demands
- List the only exponents allowed
- Take the largest exponent, b = 2
- Compare the two cases and pick the smaller sum