There is a number which has 8 divisors including, 8, itself, and 1. There is also a number which has 18 divisors, including 18, itself and 1. The difference between these numbers is 28. What are the two numbers?

If a = 2³p with p>2 and prime, then a has 8 divisors including 8, itself, and 1. Let b = (2²)(3²)q with q>3 and prime. It follows that b has 18 divisors including 18, itself, and 1. So 36q - 8p =±28 leads to p = (9q ± 7)/2. This has many solutions (probably an infinite number). For example:

p=19, q=5, forcing a=152, b=180.

p=73, q=17, forcing a=584, b=612.

p=53, q=11, forcing a=424, b=396.

p=89, q=19, forcing a=712, b=684. etc.

Note: I also found that any other form for a and b does not work.

Posted by Dennis on 2006-09-18 15:53:12