# Math Help - Abstract Algebra

1. ## Abstract Algebra

Show that Z (integers) is generated by 5 and 7.

I can sit and play with those two numbers to get all Z, but can't come up with a formula. Can someone show how to do this?

Thanks

2. ## Re: Abstract Algebra

Originally Posted by jzellt
Show that Z (integers) is generated by 5 and 7.
For all $k\in\mathbb{Z}$ we have $k=k\cdot 1=k\left(3\cdot 5+(-2)\cdot 7\right)=(3k)\cdot 5+(-2k)\cdot 7.$

3. ## Re: Abstract Algebra

Thanks! That's brilliant!

4. ## Re: Abstract Algebra

Originally Posted by jzellt
Thanks! That's brilliant!
Not very brilliant . It is a particular case of the Bezout equallity.

5. ## Re: Abstract Algebra

the subgroup of Z generated by the integers k and m is: <gcd(k,m)>. if two numbers are co-prime, they generate all of Z, since 1 generates Z (Z is cyclic).