AMC 10 · 2013 · #24
Grade 6 number-theoryPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The word "nice" hides a divisor-sum condition, so the first move is to name the primes. Tool #4 (Introduce a Variable): write m = pq and turn its divisor sum 1 + p + q + pq into the factored form (1+p)(1+q) — this converts the murky question "is n nice?" into the sharp test "can n be split into two factors that are each one-more-than-a-prime?" Tool #7 (Identify Subproblems) first splits four-divisor numbers into the two shapes p³ and pq. Tool #3 (Eliminate Possibilities) uses parity to knock out all five odd numbers at once. Tool #2 (Make a Systematic List) then tests each surviving even number through its factor pairs.
Two shapes of a four-divisor number
Exactly four divisors happens only two ways: m = p³ with divisors 1, p, p², p³, or m = pq for distinct primes with 1, p, q, pq.
The divisor count equals the product of (each exponent +1), and 4 only comes from 4 itself or 2 × 2 — that is, p³ or pq.
The divisor count is the product of each exponent plus one, so four divisors leaves only two shapes.
▸ Why?
Every number has exactly one prime recipe, so its divisors are built by choosing an exponent for each prime.
▸ Why?
Those choices are made independently for each prime, so the counts multiply.
Factor the divisor sum
For m = pq the four divisors add to 1 + p + q + pq, which factors as (1+p)(1+q) — n is nice when both factors are one more than a prime.
Grouping the four-term sum into a single product is the whole trick — it turns "is n nice?" into a factoring question.
6.EE.A.3Introduce A VariableThe cube shape overshoots
The cube sums 1 + p + p² + p³ jump from 1464 at p = 11 straight to 2380 at p = 13, skipping 2010–2019 entirely — only pq survives.
Cubes grow so fast that consecutive divisor sums jump right over the ten-number target.
6.EE.A.2Make A Systematic ListOdd numbers cannot be nice
At least one prime is odd, so a factor 1+p is even and n = (1+p)(1+q) is even — only 2010, 2012, 2014, 2016, 2018 remain.
An odd prime plus one is even, and any product containing an even factor is even.
2.OA.C.3Eliminate PossibilitiesTest the five even candidates
Need a split a · b with a-1, b-1 both prime: 2010, 2012, 2014 fail (334, 66, 502, 52 are composite) and 2018 splits only as 2 · 1009.
Running each factor pair through the "one-more-than-a-prime" test quickly exhausts every option.
4.OA.B.4Make A Systematic List2016 is the one nice number
2016 = 4 · 504 = (1+3)(1+503) with 3 and 503 prime, so m = 1509 works — exactly one of the ten is nice, choice (A).
One valid split — 3 and 503 both prime — is enough to make 2016 nice, and it is the only number in the window that admits such a split.
4.OA.B.4Eliminate PossibilitiesA number with exactly four divisors is pq, and its divisor sum factors as (1+p)(1+q) — so "nice" just means you can split n into two parts that are each one-more-than-a-prime; only 2016 = 4 · 504 = (1+3)(1+503) works, so the answer is 1, choice (A).
- Two shapes of a four-divisor number
- Factor the divisor sum
- The cube shape overshoots
- Odd numbers cannot be nice
- Test the five even candidates
- 2016 is the one nice number