Let n be a positive integer. How many different values can gcd(n+5, n+11) attain?
First this problem is equivalent to "Let m be a positive integer such that m>=6,How many different values can gcd(m,m+6) attain?"
Let's proof gcd(m,m+6)<7
Suppose p is factor of m and satisfy p>=7 and m/p=q,that m=p*q.
Then,we can write m+6 as p*q+6
(m+6)%p=(p*q+6)%p=6,that is p can not be factor of m+6
Therefore,gcd(m,m+6)<=6,acctually are the divisors of 6.
Why 4 and 5 can't be attain from gcd(m,m+6)? You can proof it by using above procedure.
1 can be gcd(m,m+6),for example 7 and 13
2 can be gcd(m,m+6),for exampe 8 and 14
3 can be gcd(m,m+6),for exampe 9 and 15
6 can be gcd(m,m+6),for exampe 6 and 12
(I am not a master on number theory.This is just a little thought on this one.By the way,is there any abstract algebra to do this?)