What are the possible equivalence classes of a square modulo 16? Prove your claim.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1797707a-0d96-41d6-8f45-19c20e886f5b%2Fb79483d2-5569-4e55-b424-c54120f628a3%2Fe47dgvm_processed.png&w=3840&q=75)
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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