Statement 1: j and k are consecutive positive integers.alex.gellatly wrote:What is the greatest common factor of the positive integers j and k?
1) k = j+1
2) jk is divisible by 5
Thanks. You made it clear. Can this logic also be applied to the integers j and j+3 and j and j+n (when n is a prime)?tutorphd wrote:The important thing to understand here is why consecutive numbers j and j+1 do not have any other common factors except -1 and 1.
Let's take a factor of j, call it n, and try to divide j+1 by it:
(j+1)/n = j/n + 1/n
j/n is integer since n is a factor of j BUT 1/n is never integer EXCEPT when n = +/- 1. The result j/n + 1/n will be integer only for n = +/- 1 which means j+1 is divisible by a factor of j only when that factor is n = +/- 1.
Similar logic can be applied to the integers j and j+2. The only common factors they will have are +/- 1 and +/- 2.
