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Magic 8's (Posted on 2006-09-18) Difficulty: 4 of 5
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?

No Solution Yet Submitted by joshua    
Rating: 4.0000 (2 votes)

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Solution Solution | Comment 5 of 8 |

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 2006-09-18 16:54:38

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