AMC 10 · 2014 · #12
Grade 6 number-theoryPick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Listing the huge divisors of N from the top down is painful. Tool #16 (Change Focus) flips the problem: every divisor d of N pairs with a partner N/d, and their product is N. As d runs through the SMALLEST divisors 1, 2, 4, 5, …, the partners N/d run through the LARGEST divisors N, N/2, N/4, … in matching order. So the fifth-largest divisor is just N divided by the fifth-smallest divisor — a much easier target. Tool #7 (Identify Subproblems) handles the setup: prime-factor N so we know exactly which small numbers divide it. Tool #2 (Systematic List) then walks the smallest divisors in order to find the fifth one, after which a single division finishes the job.
Flip largest to smallest
Divisors pair up as d · (N/d) = N, so the fifth-largest divisor is N divided by the fifth-smallest divisor.
Divisors line up in partner pairs that multiply to N, so counting down from the top mirrors counting up from the bottom.
Divisors line up in partner pairs that multiply to the number, so counting from the top mirrors counting from the bottom.
▸ Why?
Divisors come in such pairs, so the largest matches the smallest, the next matches the next.
▸ Why?
Each divisor has exactly one partner, so the two lists match up without leftovers.
Split off the powers of ten
N = 2014 × 1,000,000, and the six trailing zeros hand over 1,000,000 = 10⁶ = 2⁶ · 5⁶.
Every trailing zero is one factor of 10 = 2 × 5, so six zeros hand you 2⁶ · 5⁶ for free.
5.NBT.A.2Identify SubproblemsPrime factorize N
2014 = 2 · 19 · 53, so N = 2⁷ · 5⁶ · 19 · 53 — its only primes are 2, 5, 19, 53, never 3 or 7.
Once you know the prime bricks, you know exactly which small numbers are allowed to divide N.
6.NS.B.4Identify SubproblemsList the five smallest divisors
With no 3 and no 7 available, the divisors start 1, 2, 4, 5, 8 — the fifth-smallest is 8.
Skip any number whose primes (3 or 7) are missing from N, and the divisor ladder builds itself.
4.OA.B.4Make A Systematic ListDivide N by 8
Peeling off 2³ gives N/8 = 2⁴ · 5⁶ · 19 · 53 = 251,750,000, choice (C).
Dividing by 8 simply peels off three of the 2s in the factorization.
6.NS.B.2Identify SubproblemsThe fifth-biggest divisor is just the number divided by the fifth-smallest divisor, because divisors always come in partner pairs that multiply to the whole.
- Flip largest to smallest
- Split off the powers of ten
- Prime factorize N
- List the five smallest divisors
- Divide N by 8