# If $*$is a binary operation on set N, of natural number defined as $a*b = HCF{\text{ of (a,b)}}{\text{.}}$ Evaluate$3*\left( {2*5} \right)$.

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Hint: Let’s understand the physical interpretation of this binary operator in view of this question, if this binary operator $*$ is applied between two natural numbers than it refers to the HCF of both the numbers that is $a*b = HCF{\text{ of (a,b)}}{\text{.}}$

Complete step-by-step answer:

Now, it is given that $*$ is a binary operation on a set of natural numbers N and the definition of this operator implies that

$ \Rightarrow a*b = HCF{\text{ of (a,b)}}{\text{.}}$…………………………………… (1)

Now, we need to evaluate $3*\left( {2*5} \right)$

Firstly let’s consider the inner bracket operation first that is $\left( {2*5} \right)$

Hence, using the definition provided in equation (1) we can write it as

$ \Rightarrow 3*\left( {HCF\left( {2,5} \right)} \right)$……………………… (2)

Now, the factor of 2 is $2 \times 1$ and the factor of 5 is$1 \times 5$. Thus the highest common factors of 2 and 5 is 1.

Hence, equation (2), gets reduced to

$ \Rightarrow 3*1$

Now, again using the definition of equation (1) we can write the above as

$ \Rightarrow 3*1 = HCF\left( {3,1} \right)$……………….. (3)

Now, the factor of 3 is $3 \times 1$ and 1 doesn't have any factor hence the HCF of 1 and 3 will be 1.

Thus equation (1) gets reduced to 1.

Hence, the value of $3*\left( {2*5} \right) = 1$

Note: Whenever we face such types of problems the key point is to simplify the question part by part using the given definition of the operator. Simplifying from one side as we have done above will simplify it eventually and help to get on the right track to reach the solution. The set on which the binary operator being used also needs to be taken care of like in this case it was the set of natural numbers (N) , therefore no non-natural number is being provided in the argument of the binary operator.

Complete step-by-step answer:

Now, it is given that $*$ is a binary operation on a set of natural numbers N and the definition of this operator implies that

$ \Rightarrow a*b = HCF{\text{ of (a,b)}}{\text{.}}$…………………………………… (1)

Now, we need to evaluate $3*\left( {2*5} \right)$

Firstly let’s consider the inner bracket operation first that is $\left( {2*5} \right)$

Hence, using the definition provided in equation (1) we can write it as

$ \Rightarrow 3*\left( {HCF\left( {2,5} \right)} \right)$……………………… (2)

Now, the factor of 2 is $2 \times 1$ and the factor of 5 is$1 \times 5$. Thus the highest common factors of 2 and 5 is 1.

Hence, equation (2), gets reduced to

$ \Rightarrow 3*1$

Now, again using the definition of equation (1) we can write the above as

$ \Rightarrow 3*1 = HCF\left( {3,1} \right)$……………….. (3)

Now, the factor of 3 is $3 \times 1$ and 1 doesn't have any factor hence the HCF of 1 and 3 will be 1.

Thus equation (1) gets reduced to 1.

Hence, the value of $3*\left( {2*5} \right) = 1$

Note: Whenever we face such types of problems the key point is to simplify the question part by part using the given definition of the operator. Simplifying from one side as we have done above will simplify it eventually and help to get on the right track to reach the solution. The set on which the binary operator being used also needs to be taken care of like in this case it was the set of natural numbers (N) , therefore no non-natural number is being provided in the argument of the binary operator.

Last updated date: 20th Sep 2023

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