Concept guide
Congruence modulo a positive modulus: the remainder definition, the witness form, preservation by operations and the limits of cancellation
From the MKO's definition (n>0, a≡b (mod n) ⇔ a mod n=b mod n): (1) decide 100≡23 (mod 7); (2) prove that this is equivalent to the witness form 'there are naturals k, l with a+nk=b+nl', and give witnesses for 100 and 23; (3) explain why congruence is an equivalence relation; (4) prove that addition and multiplication preserve congruence, and from 15≡1 (mod 7) obtain 115≡24 and 1500≡23 (mod 7); (5) use 2·3≡2·0 (mod 6) to show that a common factor cannot be cancelled unconditionally; (6) explain why different moduli cannot be chained, and why the modulus 0 is excluded.