JAMB 1994 · UME · Q22

A binary operation $*$ is defined on the set of all positive integers by $a * b = ab$ . Which of the following properties does NOT hold?

Worked solution (try it first)

Closure and associativity hold: the product of two positive integers is a positive integer, and $(ab)c = a(bc)$ .
The identity is 1, because $a \times 1 = a$ , and 1 is a positive integer.
An inverse of 2 would need $2b = 1$ , so $b = \frac12$ , which is not a positive integer.
So inverse fails, option D.