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L (mod 10)=2,1,3,4,7,1,8,9,7,6,3, 9, 2,1,3,4,7,1,8,9,7,6,3, 9, 2,1,3,4,7,3…….. In this case, the first ‘9’ highlighted in red is the 12 th Lucas Number. The second 9 highlighted in red is the 24 th Lucas Number.
From the lists provided in the above sequences, of the Lucas Numbers in modular representation, we can be able to deduce that for L (mod 8), after every 12 Lucas Numbers, the 12 th number is a 9 hence, one can come into a conclusion that for L n sequence of the Lucas Numbers, if the last digit is a, then also in the sequence L n+12 has the last digit as a. for instance, from the above sequence for L (mod 8): let L n be 2,1,3,4,7,3,2,5,7,4,3,7, with seven being the last digit, then, L n+12 will be, 2,1,3,4,7,3,2,5,7,4,3,7,2,1,3,4,7,3,2,5,7,4,3, 7. The highlighted number seven is thus the last digit. This, therefore, applied for every L n+k where k is a multiple of 12.
