Problem 10 · 2007 AMC 8
Easy
Number Theory
sigma-function
For any positive integer n, define [n] to be the sum of the positive factors of n. For example, [6] = 1 + 2 + 3 + 6 = 12. Find [[11]].
Show answer
Answer: D — 28.
Show hints
Hint 1 of 2
Nested brackets mean work inside-out: settle [11] completely before touching the outer bracket. The clue is that 11 is prime — its only factors are 1 and itself.
Still stuck? Show hint 2 →
Hint 2 of 2
Factors come in pairs: list divisors as pairs that multiply to n (1&12, 2&6, 3&4) so you never miss one.
Show solution
Approach: evaluate the inner bracket first, then the outer
- Inner first: 11 is prime, so its only factors are 1 and 11 ⇒ [11] = 1 + 11 = 12.
- Now [12]. Pair up the divisors so none escape: 1·12, 2·6, 3·4 — that's {1, 2, 3, 4, 6, 12}.
- Add them: 1 + 2 + 3 + 4 + 6 + 12 = 28.
- Habit worth forming: finding divisors in multiply-to-n pairs guarantees a complete list — the same trick that makes counting factors fast.
Mark:
· log in to save