LCM is a Million

Let the two numbers be $x$ , $y$ . We know that $1,000,000 = 2^6 \cdot 5^6$ . To make the LCM, either of $x$ or $y$ must be $2^6$ and the same goes for $5^6$ . We can make 3 cases and see as below:

\[\begin{array}{|c|c|} \hline \\ \text{Power of 2 in } x & \text{Power of 2 in } y \\ \hline \\ 6& 5,4,3,2,1,0 \\ \\ 5,4,3,2,1,0&6 \\ \\ 6&6 \\ \hline

\end{array}

\implies \lbrace{\text{13 Options}}

\quad \quad

\begin{array}{|c|c|} \hline \\ \text{Power of 5 in } x & \text{Power of 5 in } y \\ \hline \\ 6& 5,4,3,2,1,0 \\ \\ 5,4,3,2,1,0&6 \\ \\ 6&6 \\ \hline

\end{array}

\implies \lbrace{\text{13 Options}}\]

Since there are 13 options to choose for power of 2, and 13 options to choose for power 5, the total options are $13 \times 13 = \boxed{169}$

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Bonus: For any given (LCM $= p_{1}^{a_{1}} \cdot p_{2}^{a_{2}} \cdots p_{n}^{a_{n}}$ ), the unique number of pairings = $\bigg(2a_{1} + 1\bigg)\cdot\bigg(2a_{2} + 1\bigg)\cdots \cdot\bigg(2a_{n} + 1\bigg)$