Problem 16 · 2019 Math Kangaroo
Hard
Number Theory
divisibilityfactor-pairs
A positive integer \(n\) is called good if its biggest factor (apart from \(n\) itself) is equal to \(n - 6\). How many good positive integers are there?
Show answer
Answer: C — 3
Show hints
Hint 1 of 2
The biggest factor of \(n\) other than \(n\) is \(n\) divided by its smallest prime factor \(p\).
Still stuck? Show hint 2 →
Hint 2 of 2
Set \(n/p = n - 6\) and solve \(n = 6p/(p-1)\), then see which primes \(p\) make \(n\) a whole number.
Show solution
Approach: the largest proper factor is \(n\) over its smallest prime factor
- The largest proper factor of \(n\) is \(n/p\), where \(p\) is the smallest prime factor of \(n\).
- Require \(n/p = n - 6\), i.e. \(n(p-1) = 6p\), so \(n = \dfrac{6p}{p-1}\).
- \(p = 2\) gives \(n = 12\), \(p = 3\) gives \(n = 9\), \(p = 7\) gives \(n = 7\) (and these each check out); no other prime makes \(n\) whole.
- So there are exactly 3 good integers — answer (C).
Mark:
· log in to save