order of QUOTIENT GROUP (Z/9Z)*(Z/8Z)*(Z/18Z)/<2,2,2>?

Answer 1

Well, the order of (Z/9Z)*(Z/8Z)*(Z/18Z) is 9*8*18 = 1296. Then we have to find the order of (2, 2, 2) in this group. We need the smallest positive integer k such that 2k is divisible by 9, 8 and 18. This will be when 2k = 72 (9 and 8 are coprime, so LCM is 9*8 = 72 which is divisible by 18). So k = 36, i.e. the order of (2, 2, 2) is 36. Thus the order of the quotient group is 1296/36 = 36.