Hi,

You posted this same question three different times. Please don't do this. Post your question only once.

What is the subgroup generated by {10,15}? It is the intersection of all subgroups that contain both 10 and 15. Let H={10m +15n : m and n are integers}. You can prove H is a subgroup. Furthermore, any subgroup containing both 10 and 15 must contain each element of H. So H is the subgroup generated by 10 and 15. Now the smallest positive integer in H is 5 (5 = (-1)*10 + 1*15) since one characterization of gcd(10,15) is the smallest positive member of H. Finally, then

H={5n : n is an integer}.