my try.....

a + 11b=13m

a + 13b=11n

solving we have a= (169m-121n)/2

and b=(11n-13m)/2

and a+b =(156m-110n)/2...........1)

now as a>0 and b>0

we have.....   11/13> m/n>121/169

or 0.85 > m/n > 0.72

we take a favourable case of 0.75to get m and n as close to each other as possible so as to get the minimum value of the expression of 1)

due to which m/n=3/4.....m=3c and n=4c.......for integral values of a+b , c shud be even and 2 being the min evn number we have m=6 and n=8

using this we have min(a+b) as (936-880)/2 = 56/2 = 28....

so the minimum value of the expression (a+b) = 28.....