LCM of Co-prime Numbers x and y

The question asks for the Least Common Multiple (LCM) of two numbers, x and y, given that they are co-prime.

What are Co-prime Numbers?

Two integers are considered co-prime (or relatively prime) if their greatest common divisor (GCD) is 1. This means the only positive integer that divides both of them is 1.

Mathematically, if x and y are co-prime, then: $ \text{GCD}(x, y) = 1 $

Relationship Between LCM, GCD, and Product

There is a fundamental relationship between the LCM and GCD of two numbers and their product. For any two positive integers x and y, the following equation holds true: $ x \times y = \text{LCM}(x, y) \times \text{GCD}(x, y) $

Calculating LCM for Co-prime Numbers

Now, let's apply this relationship to the specific case where x and y are co-prime.

1. We know that for co-prime numbers, $ \text{GCD}(x, y) = 1 $.
2. Substitute this value into the general formula: $ x \times y = \text{LCM}(x, y) \times 1 $
3. Simplifying the equation, we get: $ x \times y = \text{LCM}(x, y) $

This shows that the LCM of two co-prime numbers is simply the product of the two numbers.

Conclusion

Therefore, if x and y are co-primes, their LCM is $ \mathbf{xy} $. This matches option 2.