let a + b = y ........(1)



now,



a + 11b = 13x............(2)



and a + 13b = 11z ..............(3)



where x and z r integeres >=1



sovle above 3 equations to get



y = 78x - 55z

therefore



a + b = 78x - 55z



for min value x=z=1



so, min(a+b ) = 23