First of all we are speaking about positive natural numbers a and b right?
if so then I think the minimum is achieved by
a=23 and
b=5.
Actually we have a+2b is divisible by 11 and a-2b is divisible by 13.
if
a-2b=0 then
a=2b and
3b is divisible by 11 so
and
a+b is at least 33.if
a-2b=13 then
a+2b=13+4b is divisible by 11 so from here
and a+b is at least 28. And the last case if
is easy now
So the minimal sum is
28.
