Given that:
A + 13*B is divisible by 11 and:
A + 11*B is divisible by 13
whenever, each of A and B is a positive integer.

Determine the minimum value of A+B.

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 2013-10-16 12:56:00