Please answer both a b Thank you A zero divisor is a nonzer

Please answer both a & b. Thank you

A zero divisor is a nonzero element a of a ring R such that there exists a nonzero b R with ab = 0. a) Verify that 12 is a zero divisor in Z_18 by explicitly finding a nonzero x such that 12x = 0. b) Find all zero divisors in Z_18. (Just use the gcd test for zero divisors.)

Solution

a. Z₁₈={ 0,1,2,3,-------17}

a=12 , 12 x₁₈  b=0 then \'b\' is called zero divisor of \'12\'

when b=3 , 12x₁₈3=36 =0 (mod18) => 3 is the zero divisor of 12

when b= 6 , 12x₁₈  6 =72 =0 (mod18) => 6is the zero divisor of 12

when b= 9 , 12 x₁₈9=108 = 0(mod18) => 9 is the zero div of 12

when b=12 12 x₁₈12 = 144 =0 (mod18) => 12 is the zero div of 12

b = 15 , 12 x₁₈ 15 = 180=0 (mod18) => 15 is the zero div of 12

conclusion : zero divisors of 12 are : 3,6,9,12,15

b . 1, 5 , 7 , 11, 13, 17 are relatively prime to 18 they will not have zero div inZ₁₈

zero div of 2 and 16 ; 9

zero div of 3 and 15 : 6, 12

zero div of 4 and 14 : 9

zero div of 6 and 12 : 3, 6 ,9 ,12,15

zero div of 8and 10 : 9

zero div of 9 : 2,4, 6 ,8,10,12