LCM of two numbers a and b is (a, b), which of the following is always true?

To determine which statement is always true, we need to understand the concept of LCM (Least Common Multiple) and its properties.

Introduction:
The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all the given numbers.

Properties of LCM:
1. The LCM of any two numbers a and b is always greater than or equal to both a and b.
2. The LCM of any two numbers a and b is equal to the product of a and b divided by their greatest common divisor (GCD).
LCM(a, b) = (a * b) / GCD(a, b)

Analysis of the given options:
(a) a is prime: This statement is not always true because LCM(a, b) can be calculated for any two numbers, regardless of whether they are prime or composite. So, a does not have to be prime for the LCM to be defined.

(b) b is prime: This statement is not always true for the same reason as option (a). LCM(a, b) can be calculated for any two numbers, regardless of whether they are prime or composite.

(c) a and b are co-prime: This statement is not always true because LCM(a, b) can be calculated even if a and b are not co-prime. Co-prime numbers are those that do not have any common factors other than 1. However, the LCM is still defined for any two numbers, whether they are co-prime or not.

(d) none of these: This option is the correct answer because none of the given statements are always true. The LCM of two numbers is determined by the values of the numbers themselves, not by whether they are prime, composite, or co-prime.

Conclusion:
The correct answer is option (d) "none of these". None of the given statements are always true for the LCM of two numbers. The LCM can be calculated for any two numbers, regardless of whether they are prime, composite, or co-prime.