MathDB

In the set S_n \equal{} \{1, 2,\ldots ,n\} a new multiplication

a*b

is defined with the following properties: (i) c \equal{} a * b is in

S_n

for any

a \in S_n, b \in S_n.

(ii) If the ordinary product

a \cdot b

is less than or equal to

n,

then a*b \equal{} a \cdot b. (iii) The ordinary rules of multiplication hold for

*,

i.e.: (1) a * b \equal{} b * a (commutativity) (2) (a * b) * c \equal{} a * (b * c) (associativity) (3) If a * b \equal{} a * c then b \equal{} c (cancellation law). Find a suitable multiplication table for the new product for n \equal{} 11 and n \equal{} 12.

Problem Statement