Calculators allowed (2)...

Let x = .3650650650... ; which makes 10000x = 3650.650650650... and 10x=3.650650650... The digits behind the decimal point are identical repeating tails and 10000x - 10x = 3650 - 3 (repeating tails cancel) Simplifying the equation gives us 9990x=3647 and x has a numerator of 3647 and a denominator of 9990. The GCD of A and B is 1. Finally; $A + B = 3647 + 9990 = \boxed{13637}$

0.30650650650650650... = 0.3 + 0.0650650650650... Notice that 0.3 is not part of the repeating pattern of the decimal number. Set aside it and just focused first on 0.0650650650... part Let x be the number you want to convert to fraction. (1) x = 0.065065065065... Multiply both sides of equation (1) by 10 (2) 1000x = 65.065065065065... Combine two equations to form system-of-linear-equation-like expression: (2) 1000x = 65.065065065065... (1) x = 0.065065065065... Subtract 999x = 65 x = 65/999

Therefore 0.065065065065... = 65/999 This time, we will now add the fraction form of 0.3 which is 3/10 and add it to 65/999 to obtain same value. 0.3 + 0.065065065065... = 3/10 + 65/999 = 3647/999 Therefore 0.365065065065... = 3647/9990

As the question asked A+B = 3647 + 9990 = 13637

Observe the result...

0.365065065065...

we see that 065 repeats repeatedly.

0.365065065065... can be written as:

0.3+0.065+0.000065+0.000000065.......

=0.3+ $\frac{65}{1000}$ + $\frac{65}{1000000}$ .....

=0.3+65[ $\frac{1}{1000}$ + $\frac{1}{1000000}$ ..... ]

=Thus one part of the above equation is in G.P. where r= $\frac{1}{1000}$ , thus

$S_{∞}$ = $\frac{a}{1-r}$

= $\frac{1}{1000}$ / 1- $\frac{1}{1000}$

= $\frac{1}{999}$

continuing from where we left...

= $\frac{3}{10}$ + $\frac{65}{999}$

= $\frac{3647}{9990}$

therefore A+B=13637