AMC 10 · 2012 · #17

Grade 7 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint

difference-of-cubesgcddivisibility-rules convert-to-algebracasework ↑ Prerequisites: prime-factorization

📏 Medium solution 💡 3 insights

Problem

Let a and b be relatively prime positive integers with a larger than b, and suppose

\frac{a^3-b^3}{(a-b)^3} = \frac{73}{3}

. What is a - b?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

The answer is one of five whole numbers, so the fastest route is to turn the messy fraction into a clean divisibility fact about a - b, then throw out every choice that fails it. Factoring the difference of cubes collapses the ratio, and a short algebra step turns it into 9ab = 70(a - b)². That single equation tells us a - b must be a multiple of 3, and only one choice qualifies.

1STEP 1

Factor the difference of cubes

Factor $a^3-b^3=(a-b)(a^2+ab+b^2)$ and cancel one $(a-b)$ , leaving $\frac{a^2+ab+b^2}{(a-b)^2}$ .

(a³-b³)/((a-b)³)=(a-b)(a²+ab+b²)/((a-b)³)=(a²+ab+b²)/((a-b)²)

2STEP 2

Split off the whole-number part

Since $a^2+ab+b^2=(a-b)^2+3ab$ , the ratio splits as $1+\frac{3ab}{(a-b)^2}=\frac{73}{3}$ .

a²+ab+b²=(a-b)²+3ab → 1+3ab/((a-b)²)=73/3

3STEP 3

Clear the fractions

Subtract 1: $\frac{3ab}{(a-b)^2}=\frac{70}{3}$ . Cross-multiplying clears both threes and gives $9ab=70(a-b)^2$ .

3ab/((a-b)²)=70/3 → 9ab=70(a-b)²

4STEP 4

Force a - b to be a multiple of 3

The left side is divisible by 9 and $\gcd(70,9)=1$ , so 9 divides $(a-b)^2$ — hence 3 divides $a-b$ .

9 ∣ 70(a-b)², gcd(70,9)=1 → 9 ∣ (a-b)² → 3 ∣ (a-b)

5STEP 5

Eliminate and check the survivor

The only multiple of 3 among the choices is 3, and it works: $ab=70$ , so $a=10$ , $b=7$ with $\frac{657}{27}=\frac{73}{3}$ .

a-b=3: 9ab=70 · 9→ ab=70, a=10, b=7, gcd(10,7)=1

Answer

Plug a = 10, b = 7 back in: a³ - b³ = 1000 - 343 = 657, and (a - b)³ = 27, so the ratio is 657/27 = 73/3, exactly as required. Also a = 10 and b = 7 are relatively prime and a ＞ b ＞ 0, so every condition holds and a - b = 3 is confirmed.

💡Key takeaway

Factor the difference of cubes, boil the fraction down to one equation, and let a divisibility rule knock out every answer but the one that fits.

Factor the difference of cubes
Split off the whole-number part
Clear the fractions
Force a - b to be a multiple of 3
Eliminate and check the survivor

Factor the difference of cubes

Split off the whole-number part

Clear the fractions

Force a - b to be a multiple of 3

Eliminate and check the survivor

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