(c) Please draw Hasse diagrams for the posets (?ଵ, ?) and (?ଶ,?) where ?ଵ = {8, 168, 4, 112, 28}, ?ଶ = {9, 6, 72, 1512, 108, 3, 36} and ? is defined above. Please explain whether they are lattices or not. (10%)

6. The following is an algorithm for fast modular exponentiation: Procedure fast modular exponentiation (?: integer, ? = (?௞ିଵ?௞ିଶ … ?ଵ?଴)ଶ expressed as the binary representation, ?: positive integer) ?????? = 1; for ? = ? − 1 to 0 { ?????? = (?????? ∗ ??????) ??? ?; if (?௜ == 1) ?????? = (?????? ∗ ?) ??? ?; } return ??????; // the result equals ? ௘ mod ? (a) Please describe how to calculate ? ଼ଷ mod ? (i.e., the exponent ? = (83)ଵ଴: decimal representation) by the above algorithm and explain it is correct. (10%)

(c) Please draw Hasse diagrams for the posets (?ଵ, ?) and (?ଶ,?) where
?ଵ = {8, 168, 4, 112, 28}, ?ଶ = {9, 6, 72, 1512, 108, 3, 36} and ? is
defined above. Please explain whether they are lattices or not. (10%)

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