Question

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

Answer 1

Let * be the binary operation on ℕ defined by
a * b = H.C.F of a and b
b * a = H.C.F of b and a
we know that , H.C.F of a and b = H.C.F of b and a
e.g., a * b = b * a
therefore, * is commutative.

(1 * 2) * 3 = {H.C.F of 1 and 2 }* 3
= 1 * 3 = H.C.F of 1 and 3 = 1
1 * (2 * 3) = 1 * {H.C.F of 2 and 3 }
= 1 * 1 = H.C.F of 1 and 1 = 1
e.g., (1 * 2) * 3 = 1 * (2 * 3) = 1 , where 1, 2 , 3 ∈ ℕ.
therefore, * is associative .

Now, an element e ∈ ℕ will be the identity for the operation.
Now, if a * e = a = e * a, ∀ a ∈ ℕ.
But, this is not true for any a ∈ ℕ.
Therefore, the operation * does not have any identity in ℕ.