Congruent modulo n

Asked Jul 13 '22 at 17:03

Active Jul 13 '22 at 17:16

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Given a natural number n, and two integers a and b. Show that if:

a)a ≡ b and b≡c, it follows that a≡c

b)a≡b and c≡d, it follows that a+c≡b+d and ac≡bd

c)that a≡b if and only if a and b give the same remainder when divided by n

*I solved a) by equating k_1n with k_2n (there k_1n=a-b and k_2n = b-c). Although I am not sure how to continue with examples b) and c)...

edited Jul 13 '22 at 17:14

Bill Dubuque

asked Jul 13 '22 at 17:03

Aristarchus_

1

Please show your work. For more information about the MathSE protocol on posting questions, see this article. – user2661923 Jul 13 '22 at 17:06
For (b) see the congruence sum & product rules in the first dupe, and for (c) see the 2nd dupe. – Bill Dubuque Jul 13 '22 at 17:16
In (b) $a -b = kn$, $c-d = mn$ so $a+c-(b+d) = (k+m)n$. – Rishi Jul 13 '22 at 17:25

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