Let $a,b,c\in I^{+}$ such that $\mathrm{LCM}$ of a and b is 1000, LCM of band c is 2000 and LCM of c and a is 2000, then number of ordered triplet (a, b, c) is 

We have,  $1000=2^3\cdot 5^3$

                 $2000=2^4\cdot 5^3$

So,                  $a=2^A\cdot 5^R,b=2^B\cdot 5^S,c=2^C\cdot 5^{\mathrm{\top }}$

           $$\begin{gathered} max(A,B)=3,max(A,C)=max(B,C)=4 \\ max(R,S)=max(R,T)=max(S,T)=3 \end{gathered}$$

There are 10 ways of choosing $R, S, T.$

 1 with all 3, 3 with one 2, 3 with one 1, 3 with one 0,

 C must be 4. 

There are 7 ways of choosing $A, B.$ 

1 with both 3, 3 with A= 3, B not, 3 with B = 3 and A not.

Thus, 7 ways of choosing A, B, C have 7 -1 0 = 70 ways.