jmc

The sum of the first 20 positive even integers is $2+4+\cdots +40=2(1+2+\cdots +20)$. For all $n$, $1+2+\cdots +n=n(n+1)/2$, so $2(1+2+\cdots +20)=20\cdot 21=420$.

The sum of the first 15 positive odd integers is $1+3+\cdots +29$. The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so this sum is equal to $(1+29)/2\cdot 15=225$. The positive difference between these sums is $420-225=\boxed{195}$.