Let * be a binary operation of N given by a * b = LCM (a, b) for all a, b $\in $ N. Find the value of 5 * 7?

Hint: We start solving the problem by assigning a variable to the value of given operation 5 * 7. We solve by substituting the values in the given binary operation of the problem. We recall the definition of LCM of two prime numbers as the given numbers are prime. We use the definition and find the required value.

Complete step by step answer:
Given that we have a binary operation of N given as * which is defined as a * b = LCM (a, b) for all a, b $\in $ N. We need to find the value of 5 * 7.
Let us assume the value of 5 * 7 be ‘x’.
We have got the value of x = 5 * 7.
We have got the value of x = LCM (5, 7) -(1).
We know that LCM (Least Common multiple) of two numbers is defined as the smallest number that can be divisible by both numbers. Here, we have given values 5 and 7 which are prime. We know that the LCM of two prime numbers a and b is defined as $a\times b$. We use this result in equation (1).
We have got the value of x = $5\times 7$.
We have got the value of x = 35.
We have found the value of 5 * 7 as 35.

∴ The value of the binary operation 5 * 7 is 35.

Note: We should not confuse binary operation * with multiplication of two numbers. Since the two given numbers are prime, the calculation of the LCM has become easier. If the numbers are not prime, we should not directly multiply them. Similarly, we can expect problems with a change in the binary operation.