Question

b. Prove that if a is an odd integer, then a | b2 – 1 implies that a = (a, b – 1)(a, b + 1).

Answer 1

Answedia Given data TO prove that if a is an Odd integer, then a/6²_11 implies that a = (ab-1) (0,b)) pdoof. Suppose a divides both bth and bet albri and alb-l al(bri)-(6-1) SO which is a contradiction, as a is odd. albti ०४ a lbal SO (a,bt 1) =a 00) (ab-1-a When (a,b+ 1)=Q, ther (0,6-1)-1 a= a.l (ab+)) (ab-) Similarly (0,6-1)=a, then (0,6+1) O = al = (a,b-1) (ab+ 1) TO Observe both coses we get, a= (@b-1) (a, b+1) also if Talis factodial into p and a such that a=pl and (P.q=). TO assume that ploal and a lbti such that

a= pq | (6.1). To apply ged (6-1, a) -(6-1, pd) - p Similarly (b+10) (btl, PP. 1-4 so a=pd = (a,b-1).(0,6+1). (a,b-1) (Q,b+1)