AMC 10 · 2021 · #6

Grade 9 number-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint
prime-factorizationdifference-of-squaresprimality-testdigit-sum identify-subproblemssystematic-enumeration ↑ Prerequisites: prime-factorization
📏 Medium solution 💡 3 insights
Problem
The problem hands over one fact for free: sixteen thousand three hundred eighty-four is two to the fourteenth, so its only prime divisor is two. Now shift by one and look at the neighbour below. Break that number into primes, take the largest prime that divides it, and add up the digits of that prime.

Pick an answer.

(A)
3
(B)
7
(C)
10
(D)
16
(E)
22
How to solve
Strategy Organize Information in More Ways

Written as the five digits 16383, this number gives nothing away — factoring it by trial division would mean testing primes up to 73. So the first move is Tool #15 (Organize Information in More Ways): stop reading it as a five-digit number and read it as 2¹⁴ - 1 instead. The problem already handed over that form. Rewritten once more as 128² - 1, it becomes a difference of squares, which factors on sight. Tool #7 (Identify Subproblems) then takes over: the split produces two factors, and each one has to be pushed down to primes separately before I know which prime is largest. Tool #6 (Guess and Check) closes the last gap — testing the short list of small primes needed to certify that the biggest factor really is prime.

1STEP 1

Read the number as a power

Read it as a power minus one.

16,383 = 16,384 - 1 = 2¹⁴ - 1
2STEP 2

Turn the power into a square

The even exponent makes it a perfect square.

2¹⁴ = 2⁷ · 2 = (2⁷)² = 128², so 16,383 = 128² - 1
3STEP 3

Split the difference of squares

Split the difference of squares.

128² - 1 = (128 - 1)(128 + 1) = 127 · 129 = 16,383 ✓
4STEP 4

Break the composite factor down

Break the composite factor down.

129 = 3 · 43, 16,383 = 3 · 43 · 127
5STEP 5

Certify that 127 is prime

Certify the leftover factor is prime.

√(127) ＜ 12 → test 2, 3, 5, 7, 11 → all fail → 127 is prime
6STEP 6

Take the biggest prime and add its digits

Adding its digits gives 10.

16,383 = 3 · 43 · 127 → greatest prime = 127 → 1 + 2 + 7 = 10 → (C)
Answer
10
The factorization multiplies back correctly: 3 · 43 = 129 and 129 · 127 = 16,383, so nothing was lost or invented. Every factor listed is genuinely prime — 3 and 43 by inspection, 127 by the square-root test — so 127 really is the greatest prime divisor and not just the biggest factor found so far. The digit sum 10 appears as choice (C). The two neighbouring choices are exactly the traps: (B) 7 is the digit sum of 43, the second-largest prime, and (D) 16 is the digit sum of 16,383 read as a number rather than of its prime factor. A last sanity note: 127 = 2⁷ - 1 is a Mersenne prime, which is the standard reason this particular number is famous for being prime.
💡Key takeaway

When a number sits right next to a perfect square, write it as that square minus one — a² - 1 always splits into (a-1)(a+1), which turns a long hunt for factors into a single line.

  • Read the number as a power
  • Turn the power into a square
  • Split the difference of squares
  • Break the composite factor down
  • Certify that 127 is prime
  • Take the biggest prime and add its digits

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