Can You Read My Mind?

Since one of the outputs of the monster becomes the input of the next process, we can make a series that goes like:- $a,b,a+b,a+2b,2a+3b,3a+5b,5a+8b,8a+13b,........$ If that doesn't ring any bells, notice that if both my inputs were $1$ , this series would become $\displaystyle 1,1,2,3,5,8,13,21,.....$ .

This is very similar to the Fibonacci Sequence.

The ratio of consecutive terms, as we go ahead, quickly approaches $\dfrac{1+\sqrt{5}}{2}$ , approximately $1.618$ , also known as $\phi$ .

The $13^{th}$ term is given to us as $3935$ .

$13$ is a sufficiently large number for the ratio to be very close to $1.618$ . The $12^{th}$ term, thus, would be $\dfrac{3935}{\phi}=2432$ .

Then, it's easy stuff! We can reverse the series by taking consecutive differences, getting:- $3935,2432,1503,929,574,355,219,136,83,53,30,23,7$

Hence, the product of the numbers I took is $7 \times 23$ , which is $\large \boxed{161}$ .

89a + 144b = 3935. Notice that digit sum of 144 is 9. Digit sum of 89 is 8. basically digit sum of 89 * 7 will be 2. (8 * 9 ) digitsum = 2. Well, 89*7 = 623. Find out the rest.