
Structures  GROUPS
Hy guys .... I hope that are all fine with you ....
I had this exercises to do, and I'm having problems to solving them, can some of you,help me?
**** find the subgroup of the group (Z, +, 0) generated by the set {10, 15} .
Can you give some link for explanations, or aticles about this area?
best Regards

Re: Structures  GROUPS
Hey YoungStudent.
First consider all possible situations of 10x + 15y where x and y are whole numbers (in Z) and then consider that you need to find a minimum set that satisfies this criteria.
Hint: Consider multiples of 5 and see if you can construct every multiple of 5 with the linear combination 10x + 15y.

Re: Structures  GROUPS
Hi there.. thank you very much for your time and explanation.
So, in your hint, you talk about 5, and I presume that you get it, from here right: (1)*10 + 1*15) =5, that will be our smallest positive integer in the subgroup ?

Re: Structures  GROUPS
"{Z, +, 0}" is the group of all integers with "+" as operation and "0" as the additive identity. {10, 15} "generate" the subgroup of all integers we can get by starting with 10, 15, 10, and 15 and adding them any number of times. If we call the number of times we add 10 n and the number of times we add 15 m, we get 10n+ 15m as a formula for all such numbers. In particular, we can write this as 5(2n+ 3m) showing that all numbers in this set are multiples of 5. What about the other way? If we can show that all multiples of 5 as in this set, we are done.
Suppose 10n+ 15m= 5k for some integer k. We can immediately divide by 5 to get 2n+ 3m= k.
Start by looking at 2n+ 3m= 1. It is obviously true that m= 1, n= 1 is a solution. Multipying by k gives m= k, n= k such that 2n+ 3m= 2(k)+ 3(k)= k. That is, given any integer k, there exist integers n and m such that 2n+ 3m= k and so that 10n+ 15m= 5k.
The subgroup generated by 10 and 15 is the subgroup of all multiples of 5.