[tex]O=3+5+7+9+\cdots+105[/tex]
[tex]O=(2+1)+(4+1)+(6+1)+(8+1)+\cdots+(104+1)[/tex]
[tex]O=(2+4+6+8+\cdots+104)+n[/tex]
where [tex]n[/tex] is the numbers of 1s, which is the same as the number of terms being summed in either [tex]O[/tex] or [tex]E[/tex]. It's easy to see what that number is from [tex]E[/tex]; factoring a 2 from each term, you end up adding up the consecutive integers between 1 and 52, so [tex]n=52[/tex].
So [tex]O>E[/tex], and [tex]O=E+52\implies O-E=52[/tex].
