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Last digit number

Source: 1987 National High School Mathematics League, Exam One, Problem 7

February 24, 2020

Problem Statement

k(k>1)k(k>1) is an integer, and aa is a solution to the equation x2−kx+1=0x^2-kx+1=0. For any integer n(n>10)n(n>10), the last digit number of a2n+a−2na^{2^n}+a^{-2^n} is always 77, then the last digit number of kk is________.
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