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
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Originally Posted by jzellt Show that Z (integers) is generated by 5 and 7. For all $\displaystyle k\in\mathbb{Z}$ we have $\displaystyle k=k\cdot 1=k\left(3\cdot 5+(-2)\cdot 7\right)=(3k)\cdot 5+(-2k)\cdot 7.$
Thanks! That's brilliant!
Originally Posted by jzellt Thanks! That's brilliant! Not very brilliant . It is a particular case of the Bezout equallity.
Last edited by FernandoRevilla; Sep 20th 2012 at 01:45 AM.
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).
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