2018/012) How many pairs of numbers are there whose LCM = 600

We have $600= 2^3 * 3 * 5^2$.
For 2 number to have LCM they have to be of the form $2^a3^b4^c$ so let one number
be $2^r3^s5^t$ and aother number be $2^m3^n5^p$. and
$0 <= r < 3$, $0 <= m <=3$ and either r or m = 3
So we have 7 choices for r,m r=3, m= 0 to 3 ( 4 choices) or m =3 r = 0 to 2 ( 3 choices) (m =3, r =3
is already taken care of)
$0 <= s < 1$, $0 <= n <=1$ and either s or n = 1,
3 choices s=0 , n= 1 or s =1 , n = 1 or s =1 , n = 0
$0 <= t < 2$, $0 <= p <=2$ and either t= 2 or p= 2
So we have 5 choices for t,p  t = 2, p = 0 to 2 ( 3 choices) or p =2 t = 0 to 1 ( 2 choices)
So number of pairs = 7 * 3 * 5= 105
Out of which both are 600 is one case.
(m, n) pair is same as (n,m)
So total number of ordered pairs = $1 + \frac{105-1}{2} = 53$