1. let a; b; c; d; n belong to z with n > 0. suppose a congruent b (mod n) and c congruent d (mod n). use the definition
of congruence to
(a) prove that a + c congruent b + d (mod n).
(b) prove that ac congruent bd (mod n).
expert answer

Proofs are in the explantion.

Step-by-step explanation:

We are given the following:

1) for integer .

1) for integer .

a)

Proof:

We want to show .

So we have the two equations:

a-b=kn and c-d=mn and we want to show for some integer r that we have

(a+c)-(b+d)=rn. If we do that we would have shown that .

kn+mn   =  (a-b)+(c-d)

(k+m)n   =   a-b+ c-d

(k+m)n   =   (a+c)+(-b-d)

(k+m)n  =    (a+c)-(b+d)

k+m is is just an integer

So we found integer r such that (a+c)-(b+d)=rn.

Therefore, .

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b) Proof:

We want to show .

So we have the two equations:

a-b=kn and c-d=mn and we want to show for some integer r that we have

(ac)-(bd)=tn. If we do that we would have shown that .

If a-b=kn, then a=b+kn.

If c-d=mn, then c=d+mn.

ac-bd  =  (b+kn)(d+mn)-bd

          =    bd+bmn+dkn+kmn^2-bd

          =           bmn+dkn+kmn^2

          =            n(bm+dk+kmn)

So the integer t such that (ac)-(bd)=tn is bm+dk+kmn.  

Therefore, .

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step-by-step explanation: