Problem 17 · 2020 AMC 8
Medium
Number Theory
factorizationdivisibility
How many factors of 2020 have more than 3 factors? (As an example, 12 has 6 factors, namely 1, 2, 3, 4, 6, and 12.)
Show answer
Answer: B — 7 factors.
Show hints
Hint 1 of 2
“More than 3 factors” is hard to spot one by one. Flip it: which numbers have 3 or fewer factors? There are only three types — and they're easy to recognize.
Still stuck? Show hint 2 →
Hint 2 of 2
A number has ≤3 factors only if it's 1 (one factor), a prime (two), or a prime squared (three). Count those among the divisors of 2020, then subtract from the total.
Show solution
Approach: complementary — subtract the three ‘small’ types
- 2020 = 22 · 5 · 101, so the total factor count is (2+1)(1+1)(1+1) = 12.
- The divisors with ≤3 factors fall into exactly three recognizable types: 1 (one factor); the primes 2, 5, 101 (two each); and the prime square 4 = 22 (three). That's 1 + 3 + 1 = 5.
- Remaining: 12 − 5 = 7 divisors with more than 3 factors.
- You'll see this again as: the number of divisors a number has is fixed by its prime exponents — exactly 1 factor ⇒ the number is 1; exactly 2 ⇒ prime; exactly 3 ⇒ prime squared. Memorize those three and ‘count the small ones, subtract’ becomes routine.
Mark:
· log in to save