Given the language L { w | w has unequal number of 0's and 1's } show that it's not regular.
I tried to use pumping lemma choosing the word of the form $0^p$ $1^{2p}$ ensuring that we'll have to pump zeroes. But i don't understand how can we ensure that we'll get equal number of 0's and 1's this way. Maybe i need to choose other kind of word?
Try $0^p1^n$ with some larger $n$ instead. What we want is that for all $k$ with $1\le k\le p$, there exists $m$ such that $0^{km}0^{p-k}1^n\notin L$. So we want to find $n$ such that $km+p-k=n$ for suitable $m$. In other words, $n-p$ must be a multiple fo all $k\le p$. Try $n=p!+p$.