IMC 2014 Q7
7. Just one positive integer has exactly 8 factors including 6 and 15.
What is the integer?
- A 21
- B 30
- C 45
- D 60
- E 90
Intermediate Mathematical Challenge 2014, Question 7. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 30
Since 6 and 15 are factors of the integer, its prime factors will include 2, 3 and 5. So 10 and 30 will also be factors of the required integer. Seven of its factors are now known and as 1 must also be a factor, the required integer is 30, the factors of which are 1, 2, 3, 5, 6, 10, 15, 30.
(Positive integers with exactly 8 factors are of the form \(pqr\) or \(pq^3\) or \(p^7\) where \(p\), \(q\), \(r\) are distinct primes.)
Solution © UK Mathematics Trust, from the IMC 2014 solutions. Longer versions with investigations: Solutions and investigations (PDF).