( 8 points) For

n

an integer

n>1

and integers

a

and

b

, define

ab

mod

n

if

n

divides

a-b

. (By definition,

n

divides

m

if there exists an integer

k

such that

m=nk

.) This relation,

is an equivalence relation on

Z

. Now, show that

is a congruence, that is, show that if

abmodn

and

cdmodn

, then\ a.

a+cb+dmodn

, and

5. ( 8 points) For n an integer n>1 and integers a and b, define ab mod n if n divides a - b. (By definition, n divides m if there exists an integer k such that m=nk.) This relation, is an equivalence relation on Z. Now, show that is a congruence, that is, show that if abmodn and cdmodn, then a. a+cb+dmodn, and