# Recurrences and Divisors

**Discrete Mathematics**Level 5

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 .