Question

OD (2) Show that if m>gedlm,a)>l, then [a] is a zero-divisor' in 2/m2 An element [a] in 2/m2 is a zero-divisor if [a] * [O] and there is a [b]+[O] in 2/m2 such that [a][b]=[0]. (3) Which elements in Z/m2 have multiplicative inverses? Hint: If d=acdm.a), then abed (mod m), for some b€2

Answer 1

25 Here god cm, a) > let K be gcd cm, as Then m K and non-zero a K both are a elements of z mz divides a and b both then (: gcd carb) m (%) = [0] (Hae paz, am = 0 (m) = [o] (: 2 15 commutative) =) m2 Here, m # [o] anel [a] + [0] mz Z - m2 Hence, [caz is left zero divisor. in z similarly, com la Boog (am is cummuterie) cha) (vatta am=c) Here, tu and [a] to Hence [a [a] is right zero divisor in z Hence [a] is a zero divisor in I a = [0] m K mz m2

nz mz non 2 mz 3) we daim that the elements who ma arall which have god cm, c) = 1 ase the elements which have multiplicative inverses in In (2) we have proved that it [a] is res divisur in when gcd cm, a) > 1 Flence, "a" is not does not have multiplicative inverse when gcdcm,a) 211 How, when gcdcm, a) = 1 Thes 13y Fuclidean algorithm, 3 K1, K2 Stock EZ Sulthoot, such that, mk, takz= But ma = 0,, V QER mz ma Hence, mkit akz=1 =) Ot ak2 = 1 =) ak2=1 K2 is multiplicative invesse ot" [a). Hence lea" ET has multiplicative inverse m 71 in pe z mz if .gcd cm, a) = 1)