Question

Make a conjecture: In the ring (Z_n, +, \times), the element k is a unit if and only if

$INSTRUCTIONS Make a conjecture by filling in the blank: In the ring (Z_n, +, \times), the element k is a zero divisor if and$

Answer 1

In the ring (Zn, to), the element kis a unit if and only of there exists XE Zn such that x.k = 1 In the ring (Zn, t;') the element kis a zero divisor ef and only if there exists Xe Zn such that Xk=0. 6097 427 119 427x 14+119 !1983 + 70 = 70 X1 +49. 70 49x1 + 21 49 = 21x2 +7 = 7x3 + 0 :i f.C.d(6097,427) = 7. A: If gcd (h, k) is equal to d, then k is not a zero divisor in Zn. Bi If k is not a zero divisor in Zn, then ged(n, k) is 21 . equal to 1. 6: k is a zero divisor in Znif and only if ged (b,k). is not equal tol.