The number of ordered pairs of positive integers $(a,b)$ such that LCM of a and b is $2^{3}5^{7}11^{13}$ I started by taking two numbers such as $2^{2}5^{7}11^{13}$ and $2^{3}5^{7}11^{13}$. The LCM of those two numbers is $2^{3}5^{7}11^{13}$. Similarly, If I take two numbers like $2^{3-x}5^{7-y}11^{13-z}$ and $2^{3}5^{7}11^{13}$, the LCM is $2^{3}5^{7}11^{13}$. Here $0\le x\le3$, $0\le y\le7$, $0\le z\le13$
The number of ways of choosing 3 numbers $x,y,z$ is $^4C_1\cdot^8C_1\cdot^{14}C_1$
The above value is only for $b=2^{3}5^{7}11^{13}$.But now I have to consider another value of $b$ and the the solution becomes lengthy. Is this the correct approach?

Hint: 


*

*One of $a$ and $b$ must have 3 factors two, the other less.  

*One of $a$ and $b$ must have 7 factors five, the other less.

*One of $a$ and $b$ must have 13 factors eleven, the other less.

*No other prime factors.

