AMC 10 · 2007 · #17

Grade 8 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint
prime-factorizationexponentsmultiples identify-subproblems ↑ Prerequisites: prime-factorization
📏 Medium solution 💡 2 insights
Problem
You want two positive whole numbers m and n with 75m = n³, meaning 75 times m lands exactly on a perfect cube. Among all such pairs, find the smallest possible value of m + n.

Pick an answer.

(A)
15
(B)
30
(C)
50
(D)
60
(E)
5700

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The words "minimum possible value" point straight at Tool #14 (Extreme Principle): we push m and n down to the smallest sizes the cube condition allows. To see what that condition really is, Tool #7 (Identify Subproblems) says: first break 75 into primes, then handle each prime's exponent on its own. Tool #4 (Introduce a Variable) writes n = 3^x · 5^y so the cube n³ has exponents 3x and 3y we can control. Matching those tripled exponents against what 75 already supplies tells us the smallest legal x and y, which pins down the smallest n, then the smallest m, then their sum.

1STEP 1

Break 75 into primes

Factor 75 into primes: 75 = 3 · 25 = 3 · 5². One 3 and two 5s — those two exponents decide everything.

75 = 3¹ · 5²
2STEP 2

Write the cube in prime form

Since 75 holds only 3s and 5s, so does n. Write n = 3^x · 5^y, so n³ = 3³x · 5³y — every exponent a multiple of 3.

n = 3^x · 5^y → n³ = 3³x · 5³y
3STEP 3

Push the exponents to their minimum

For n³ to be a multiple of 75 we need 3x ≥ 1 and 3y ≥ 2, so x = y = 1: the smallest n is 15 and n³ = 3375.

3x ≥ 1, 3y ≥ 2 → x = y = 1 → n = 15, n³ = 15³ = 3375
4STEP 4

Recover m and add

Then m = 3375 ÷ 75 = 45, and since n is already minimal so is the pair: m + n = 45 + 15 = 60, choice (D).

m = 3375/75 = 45 → m + n = 45 + 15 = 60 → (D)
Answer
60
Check the pair really works: 75 · 45 = 3375 and 15³ = 15 · 15 · 15 = 225 · 15 = 3375, so 75m = n³ holds exactly. It is also genuinely the minimum: any smaller cube multiple of 75 would need exponents below 3 on the 3 or the 5, but a cube can't have an exponent of 1 or 2, so 15³ is the first cube that 75 divides. The answer 60 sits sensibly among the choices: (A) 15 and (B) 30 are too small (even n = 15 alone already uses up 15, and m can't be 0), while (E) 5700 comes from wastefully overshooting the exponents.
💡Key takeaway

To make a product a perfect cube, factor into primes and bump each exponent up to the nearest multiple of three, no higher.

  • Break 75 into primes
  • Write the cube in prime form
  • Push the exponents to their minimum
  • Recover m and add

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