Problem 24 · 2009 Math Kangaroo
Stretch
Number Theory
factorizationfactor-pairs
All factors of a number N (with the exception of 1 and N itself) are written down one after another. It turns out that the biggest of these factors is 45 times as big as the smallest. For how many numbers N is this true?
Show answer
Answer: C — 2
Show hints
Hint 1 of 2
The smallest proper factor is the least prime p; the largest proper factor is N/p.
Still stuck? Show hint 2 →
Hint 2 of 2
Set N/p = 45p, so N = 45p², and require p to really be the smallest prime factor.
Show solution
Approach: express N through its smallest and largest proper factors
- The smallest proper factor is the least prime p of N and the largest is N/p.
- 'Largest = 45 × smallest' means N/p = 45p, so N = 45p².
- Since 45 = 3²·5, p must be 2 or 3 for p to stay the smallest prime: N = 180 or N = 405.
- Both work (factors of 180 run 2…90, of 405 run 3…135), so there are 2 such numbers.
Mark:
· log in to save