AMC 10 · 2003 · #14
Grade 8 number-theoryGiven that 38⋅52=ab, where both a and b are positive integers, find the smallest possible value for a+b.
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: The number $3^8 \cdot 5^2$ is to be written as a single power $a^b$, where $a$ and $b$ are positive integers. Among all the ways to do this, find the smallest possible value of $a + b$.
Givens: The number equals $3^8 \cdot 5^2$; It must be rewritten in the form $a^b$ with $a$ and $b$ positive integers; Answer choices: (A) $25$, (B) $34$, (C) $351$, (D) $407$, (E) $900$
Unknowns: The smallest possible value of $a + b$ over all valid ways to write the number as $a^b$
Understand
Restated: The number $3^8 \cdot 5^2$ is to be written as a single power $a^b$, where $a$ and $b$ are positive integers. Among all the ways to do this, find the smallest possible value of $a + b$.
Givens: The number equals $3^8 \cdot 5^2$; It must be rewritten in the form $a^b$ with $a$ and $b$ positive integers; Answer choices: (A) $25$, (B) $34$, (C) $351$, (D) $407$, (E) $900$
Plan
Primary tool: #14 Extreme Principle
Secondary: #4 Introduce a Variable, #3 Eliminate Possibilities
The number is already in prime-factored form $3^8 \cdot 5^2$, 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.
Execute — Answer: D
6.EE.A.1 Step 1 See what "a to the b" demands
- Writing $3^8 \cdot 5^2 = a^b$ means every prime factor is split into $b$ equal groups.
- So the exponent of $3$, which is $8$, must be split evenly into $b$ groups, and the exponent of $5$, which is $2$, must also be split evenly into $b$ groups.
- That forces $b$ to 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.
6.NS.B.4 Step 2 List the only exponents allowed
- The values of $b$ that divide both $8$ and $2$ are exactly the common divisors of $8$ and $2$.
- Their greatest common factor is $2$, so the common divisors are $1$ and $2$.
- That means $b$ can only be $1$ or $2$ — there are just two cases to check.
💡 An exponent that must divide both $8$ and $2$ can be no larger than their greatest common factor.
8.EE.A.1 Step 3 Take the largest exponent, b = 2
- Choosing $b = 2$ splits each prime's exponent in half: the eight $3$s become two groups of four, and the two $5$s become two groups of one.
- So the base is $a = 3^4 \cdot 5^1 = 81 \cdot 5 = 405$.
- Squaring it recovers the original number, $405^2 = (3^4 \cdot 5)^2 = 3^8 \cdot 5^2$, so this is valid.
💡 Halving every exponent turns the number into a perfect square whose root is $3^4 \cdot 5 = 405$.
6.EE.A.1 Step 4 Compare the two cases and pick the smaller sum
- With $b = 2$: $a = 405$, so $a + b = 405 + 2 = 407$.
- With $b = 1$: $a$ is the whole number $3^8 \cdot 5^2 = 164025$, so $a + b = 164026$, which is enormous.
- The larger exponent shrinks the base far more than it adds to $b$, so $b = 2$ wins.
- The smallest possible value of $a + b$ is $407$, which is 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.1 Writing $3^8 \cdot 5^2 = a^b$ means every prime factor is split into $b$ equal g 6.NS.B.4 The values of $b$ that divide both $8$ and $2$ are exactly the common divisors o 8.EE.A.1 Choosing $b = 2$ splits each prime's exponent in half: the eight $3$s become two 6.EE.A.1 With $b = 2$: $a = 405$, so $a + b = 405 + 2 = 407$. With $b = 1$: $a$ is the wh Review
Reasonableness: Check the winning form directly: $405^2 = 164025$ and $3^8 \cdot 5^2 = 6561 \cdot 25 = 164025$, so $405^2$ really equals the original number and $a + b = 407$. The only other legal form, $b = 1$, gives $a + b = 164026$, far bigger, so $407$ is genuinely the minimum. Choices (A) $25$, (B) $34$, and (C) $351$ are all below $407$, but none corresponds to a legal $a^b$: no valid split of the exponents produces a base and exponent summing to those, since $b$ can only be $1$ or $2$. So $407$ stands.
Alternative: Reason from perfect powers instead of divisors. $3^8 \cdot 5^2$ is a perfect square because both exponents are even; its square root is $\sqrt{3^8 \cdot 5^2} = 3^4 \cdot 5 = 405$, giving $405^2$ and $a + b = 407$. It is not a perfect cube, fourth power, or any higher power, because the exponent of $5$ is only $2$ and no integer $b > 2$ divides $2$. So squaring is the strongest compression available, and $407$ is the smallest sum.
CCSS standards used (min grade 8)
6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Reading $3^8 \cdot 5^2 = a^b$ as splitting each prime's exponent into $b$ equal groups, and evaluating $3^4 \cdot 5 = 405$ and the resulting sums.)6.NS.B.4Find the greatest common factor of two whole numbers (Using $\gcd(8, 2) = 2$ to conclude the exponent $b$ can only be $1$ or $2$.)8.EE.A.1Know and apply the properties of integer exponents to generate equivalent numerical expressions (Verifying $(3^4 \cdot 5)^2 = 3^8 \cdot 5^2$ so that $405^2$ really equals the original number.)
⭐ To 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.
⭐ To 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.
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