Question

Let * be the binary operation of N defined by a * b = H.C.F. of a and b. Is * commutative? Is * associative? Does their exist identity for this operation on N?

Answer 1

Binary operation on set N is defined as
a * b = H.C.F. of a and b.

(a) We know that H.C.F. of a, b = H.C.F. of b, a.
∴ a * b = b * a
Binary operation * is commutative.

(b) a * (b * c) = a * (H.C.F. of b, c)
= H.C.F. of a and (H.C.F. of b, c)
= H.C.F. of a, b and c.
Similarly, (a * b) * c = H.C.F. of a, b and c
⇒ (a * b) * c – a * (b * c)
So, binary operation * as defined above is associative.

(c) 1 * a = a * 1 = 1 * a.
There does not exists any identity element.