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?
There appear to a number of solutions.
For the number to have exactly 8 different factors to include 1 and 8, the number must be composed of 1 and the primes 2, 2, 2 and x where x is not 2.
For the second number to have exactly 18 different factors to include 1 and 18, the number must be composed of 1 and the primes 2, 2, 3, 3 and y where y in not 2 or 3.
As the difference between the numbers is 28, the numbers should satisfy the equation [1*2*2*3*3]*y + 28 = [1*2*2*2]*x. Simplified and solving for x (and y), the equation can be written: x = (9y + 7)/2.
Seven pair of primes (x, y) that satisfy this equation are:
(19, 5), (53, 11), (73, 17), (19, 89), (23, 107), (163, 37) and (271, 61). There are more, but I do not wish to check every prime.
Edited on September 18, 2006, 5:06 pm

Posted by Dej Mar
on 20060918 16:54:38 