How many factors of $N$ are a multiple of $K$?

Question

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

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Any approach to attempt such questions ?

Answer 1

This the same as asking how many divisors for the number $$\frac NK=2^8\cdot3^7\cdot 5^5\cdot 7^4.$$

Answer 2

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

Answer 3

Hint:

Since $\;N=2^{38}\cdot3^{27}\cdot5^{15}\cdot7^{14}\;,\;\;K=2^{30}\cdot3^{20}\cdot5^{10}\cdot7^{10}\;$

You need that each multiple has at least the same ammount of prime factors as $\;K\;$ has...