How many factors of $N$ are a multiple of $K$? How many factors of $N = 12^{12} \times 14^{14} \times 15^{15}$ are a multiple of $K = 12^{10} \times 14^{10} \times 15^{10}$ ?

Any approach to attempt such questions ?

Look at it this way:
$$N=2^{38}3^{27} 5^{15} 7^{14}$$
Now ask yourself how many factors are a multiple of:
$$K=2^{30}3^{20} 5^{10} 7^{10}$$$$
