A binary operation `""^(ast)` defined on N, is given by `a^(ast)b=`H.C.F. (a, b), `foralla, b in N`. Check the commutativity and associativity.

Answer

To solve the problem, we need to check the properties of commutativity and associativity for the binary operation defined as \( a \ast b = \text{H.C.F.}(a, b) \) for all \( a, b \in \mathbb{N} \).

### Step 1: Check Commutativity

1. **Definition of Commutativity**: An operation \( \ast \) is commutative if \( a \ast b = b \ast a \) for all \( a, b \in \mathbb{N} \).

2. **Calculate \( a \ast b \)**:
\[ a \ast b = \text{H.C.F.}(a, b) \]

3. **Calculate \( b \ast a \)**:
\[ b \ast a = \text{H.C.F.}(b, a) \]

4. **Compare \( a \ast b \) and \( b \ast a \)**:
Since the H.C.F. of two numbers does not depend on the order of the numbers, we have:
\[ \text{H.C.F.}(a, b) = \text{H.C.F.}(b, a) \]

5. **Conclusion**: Therefore, \( a \ast b = b \ast a \), which confirms that the operation is commutative.

### Step 2: Check Associativity

1. **Definition of Associativity**: An operation \( \ast \) is associative if \( (a \ast b) \ast c = a \ast (b \ast c) \) for all \( a, b, c \in \mathbb{N} \).

2. **Calculate \( (a \ast b) \ast c \)**:
\[ (a \ast b) \ast c = \text{H.C.F.}(\text{H.C.F.}(a, b), c) \]

3. **Calculate \( a \ast (b \ast c) \)**:
\[ a \ast (b \ast c) = \text{H.C.F.}(a, \text{H.C.F.}(b, c)) \]

4. **Use the property of H.C.F.**:
By the property of H.C.F., we know that:
\[ \text{H.C.F.}(\text{H.C.F.}(a, b), c) = \text{H.C.F.}(a, b, c) \]
and
\[ \text{H.C.F.}(a, \text{H.C.F.}(b, c)) = \text{H.C.F.}(a, b, c) \]

5. **Compare both results**:
\[ (a \ast b) \ast c = \text{H.C.F.}(a, b, c) \]
\[ a \ast (b \ast c) = \text{H.C.F.}(a, b, c) \]

6. **Conclusion**: Since both expressions are equal, we conclude that \( (a \ast b) \ast c = a \ast (b \ast c) \), confirming that the operation is associative.

### Final Conclusion
The binary operation \( \ast \) defined by \( a \ast b = \text{H.C.F.}(a, b) \) is both commutative and associative.