LCM ! Not so easy .

First, prime-factorize $700$ . $700=5^2 \cdot 2^2 \cdot 7^1$

Now, let $a=5^{a_1} \cdot 2^{a_2} \cdot 7^{a_3}$ and $b=5^{b_1} \cdot 2^{b_2} \cdot 7^{b_3}$ .

Note that for $\text{LCM}(a,b)=700$ , $\max\{a_i,b_i\}$ has to be equal to the power of the corresponding prime factor in the prime factorization of $700$ . Let us call that power $p_i$ .

So, $\max\{a_i,b_i\}=p_i$ and $p_1=2, \ p_2=2, \ p_3=1$ .

Now as we need ordered pairs $(a,b)$ , we can have the following three cases for $i=1,2,3$ -

• $a_i>b_i, \ \max\{a_i,b_i\}=a_i$

• $b_i>a_i, \ \max\{a_i,b_i\}=b_i$

• $a_i=b_i$

The first two can be achieved in $p_i$ ways each, and the last can be achieved in only one way. Hence, the total number of ways of having $\max\{a_i,b_i\}=p_i$ is $2p_i+1$ .

Therefore, by rule of product, the number of ordered pairs is $\prod_{i=1}^{3} (2p_i+1)=\boxed{75}$