a+b=43 is the minimum value.
Let c=11*13 = 143
Then the pair: c+11*c, c+13*c or {1716, 2002} satisfies the stipulation, when a=b=143. So a 'minimum value' if such exists, must be less than 286. Using WolframAlpha,
(a+13b) = 11m, (a+11b) = 13n, 0<a<b, (a+b)<286 quickly produces a list that includes: {a = 13, b = 65, m = 78, n = 56}
Using (a+13b) = 11m, (a+11b) = 13n, 0<a<b, (a+b)<78 confirms that {a = 20, b = 23, m = 29, n = 21} is a minimum.
Note: The above fails to take into account that b might be less than a, see succeeding posts.
Edited on October 17, 2013, 3:32 am

Posted by broll
on 20131016 12:56:00 