The LCM of 96,144 and N is 576. If their HCF is 48, then a possible value of N is

Factors of the number are

$$576 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^6 \times 3^2$$

$$96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^5 \times 3$$

$$144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2$$

LCM of 96,144 and n is 576.

i.e.

$$n = 2^6 \times k = 64k$$
....(1)

HCF is 48.

$$HCF(96, 144, 64k) = 2^4 \times 3$$

$$48 = 2^4 \times 3$$

This only happen if k=3

So, substituting k=3 in (1),

$$n=64×3=192.$$

D is correct choice.