IMO 1989 LL AUS1
In the set Sn = {1, 2, . . ., n} a new multiplication a∗b is defined
IMO 1989 LL AUS1
Origin: AUS
Problem
In the set Sn = {1, 2, . . ., n} a new multiplication a∗b is defined with the following properties: (i) c = a ∗b is in Sn for any a \inSn, b \inSn. (ii) If the ordinary product a\cdotb is less than or equal to n, then a∗b = a\cdotb. (iii) The ordinary rules of multiplication hold for ∗, i.e., (1) a ∗b = b ∗a (commutativity) (2) (a ∗b) ∗c = a ∗(b ∗c) (associativity) (3) If a ∗b = a ∗c then b = c (cancellation law). Find a suitable multiplication table for the new product for n = 11 and n = 12.