Let * be the binary operation on N defined by, a * b = H.C.F. of … - Vidyadip

Question

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

✓

Answer

The binary operation * on N is defined as:
a * b = H.C.F. of a and b.
it is known that:
H.C.F. of a and b = H.C.F. of b and a. $\text{a, b}\in\text{N}$.
Therefore, a * b = b * a
Thus, the operation * is commutative.
For $\text{a, b, c}\in\text{N}$, we have:
(a * b) * c = (H.C.F of a and b) * c = H.C.F. of a, b and c
a * (b * c) = a * (H.C.F. of b and c) = H.C.F. of a, b and c
Therefore, (a * b) * c = a * (b * c)
Thus, the operation * is associative.
Now, an element $\text{e}\in\text{N}$ will be the identity for the operation
if a * e = a = e * a, $\forall\text{ a}\in\text{N}$.
But this relation is not true for any $\text{a}\in\text{N}$.
Thus, the operation * does not have any identity in N.

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