3. Prove or disprove the following statements by using the definitions of "congruence modulo n" and "divides." (a) For all positive integers a, x and y, if (a + x) = (a + y) (mod 12), then x = y (mod 12). (b) For all positive integers, a, x and y, if ax = ay (mod 12), then x = y(mod 12). (c) There exists a positive integer a > 1 so that for all x, y € Z, if ax ay (mod 12), then x = y (mod 12). (d) For all positive integers a and b, if a² = 6² (mod 12), then a = b (mod 12).
3. Prove or disprove the following statements by using the definitions of "congruence modulo n" and "divides." (a) For all positive integers a, x and y, if (a + x) = (a + y) (mod 12), then x = y (mod 12). (b) For all positive integers, a, x and y, if ax = ay (mod 12), then x = y(mod 12). (c) There exists a positive integer a > 1 so that for all x, y € Z, if ax ay (mod 12), then x = y (mod 12). (d) For all positive integers a and b, if a² = 6² (mod 12), then a = b (mod 12).
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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Transcribed Image Text:3. Prove or disprove the following statements by using the definitions of "congruence modulo
n" and "divides."
(a) For all positive integers a, x and y, if (a + x) = (a + y) (mod 12), then x = y (mod 12).
(b) For all positive integers, a, x and y, if ax = ay (mod 12), then x = y (mod 12).
(c) There exists a positive integer a > 1 so that for all x, y EZ, if ax = ay (mod 12), then
x = y (mod 12).
(d) For all positive integers a and b, if a² = 6² (mod 12), then a = b (mod 12).
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