Recurrences and Divisors

Given that \(\mathrm{a_0 = b_0 = 0}\)
\(\mathrm{a_i , b_i \geq 0} \) .......... \(\forall i\geq 0\)
\(\mathrm{a_1 : b_1 = 6:61}\)
\(\mathrm{b_1^2-a_1^2 = 3685}\)
\(\mathrm{a_n = 8a_{n-1} - 7a_{n-2}}\)
\(b_n = 65b_{n-1}- 126 b_{n-2}\)

Find the Greatest Common Divisor of

the numbers formed by last 5 digits of \(\mathrm{\text{a}_{2560}}\) and \(\mathrm{b_{1000}}\)

Details And Assumptions :-

The last five digits are taken in order as they appear i.e.

number formed by last 5 digits of the number 67865783658 is 83658 .

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