Yes . Option B cannot be the H.C.F of those numbers because according to theorem H.C.F of two numbers divides the LCM of those two particular numbers completely.In the above question all the options except B that is A,C and D divides the LCM( 1200 ) completely whereas option B does not divide the LCM completely. Therefore , the two particular numbers having 1200 as their L.C.M cannot have 500 as their HCF.

To find the HCF of two numbers, we need to first find the prime factorization of both numbers. Then, we take the common prime factors of both numbers and multiply them together to find the HCF.

Given that the LCM of the two numbers is 1200, we can say that both numbers have 1200 as a multiple. So, let's consider the prime factorization of 1200.

Prime factorization of 1200:
1200 = 2^4 * 3 * 5^2

Now, let's consider the options one by one and check if the given number can be the HCF.

a) 600:
Prime factorization of 600:
600 = 2^3 * 3 * 5^2

The common prime factors between 1200 and 600 are 2^3, 3, and 5^2. So, the HCF can be 2^3 * 3 * 5^2 = 600.

b) 500:
Prime factorization of 500:
500 = 2^2 * 5^3

The common prime factors between 1200 and 500 are 2^2 and 5. So, the HCF can be 2^2 * 5 = 20.

c) 400:
Prime factorization of 400:
400 = 2^4 * 5^2

The common prime factors between 1200 and 400 are 2^4 and 5^2. So, the HCF can be 2^4 * 5^2 = 400.

d) 200:
Prime factorization of 200:
200 = 2^3 * 5^2

The common prime factors between 1200 and 200 are 2^3 and 5^2. So, the HCF can be 2^3 * 5^2 = 200.

From the above analysis, we can see that all the given options can be the HCF except for option b) 500. Therefore, the correct answer is option b) 500.