# Abstract Algebra

• Sep 19th 2012, 11:45 PM
jzellt
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
• Sep 20th 2012, 01:31 AM
FernandoRevilla
Re: Abstract Algebra
Quote:

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

For all $\displaystyle k\in\mathbb{Z}$ we have $\displaystyle k=k\cdot 1=k\left(3\cdot 5+(-2)\cdot 7\right)=(3k)\cdot 5+(-2k)\cdot 7.$
• Sep 20th 2012, 01:34 AM
jzellt
Re: Abstract Algebra
Thanks! That's brilliant!
• Sep 20th 2012, 01:39 AM
FernandoRevilla
Re: Abstract Algebra
Quote:

Originally Posted by jzellt
Thanks! That's brilliant!

Not very brilliant :). It is a particular case of the Bezout equallity.
• Sep 20th 2012, 05:27 AM
Deveno
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).