Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
Squares are controlled modulo 16. Preamble. Recall the equivalence class, [a], of an integer, a, modulo an integer, b, is the set of all integers c satisfying: ca (modulo b), read "c is equivalent to a modulo b", i.e. blc-a, read "b divides c-a", i.e. there exists an integer n satisfying: bn=c-a, (3) read "c-a is divisible by b", e.g. the equivalence class, [3], of 3 modulo 16 is the set of all integers c which are equivalent to 3 modulo 16, i.e. the set (1) (2) ..,-13, 3, 19, 35, .... {cZ such that there exists an integer n satisfying 16n=c-3, i.e. c = Intuitively, these are integers whose remainder is 3 upon division by 16, e.g. 16n+3}. End preamble. What are the possible equivalence classes of a square modulo 16? Prove your claim.
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Transcribed Image Text:Squares are controlled modulo 16. Preamble. Recall the equivalence class, [a], of an integer, a, modulo an integer, b, is the set of all integers c satisfying: ca (modulo b), read "c is equivalent to a modulo b", i.e. blc-a, read "b divides c-a", i.e. there exists an integer n satisfying: bn=c-a, (3) read "c-a is divisible by b", e.g. the equivalence class, [3], of 3 modulo 16 is the set of all integers c which are equivalent to 3 modulo 16, i.e. the set (1) (2) ..,-13, 3, 19, 35, .... {cZ such that there exists an integer n satisfying 16n=c-3, i.e. c = Intuitively, these are integers whose remainder is 3 upon division by 16, e.g. 16n+3}. End preamble. What are the possible equivalence classes of a square modulo 16? Prove your claim.
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