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Q.
The number of ordered pairs $(x, y)$ such that $L C M$ of $x$ and $y$ is $2520$ is
Permutations and Combinations
Solution:
$2520=2^{3} \times 3^{2} \times 5 \times 7$
Ways to distribute $2^{3}$ to $(x, y)$ are:
$\left(2^{3}, 1\right),\left(2^{3}, 2\right),\left(2^{3}, 2^{2}\right),\left(2^{3}, 2^{3}\right),\left(1,2^{3}\right),\left(2,2^{3}\right),\left(2^{2}, 2^{3}\right)$
$\therefore $ Number of ways of distribution $=2 \times 3+1=7$
Similarly, other factors can also be distributed to $x$ and $y$.
Thus, the total number of ways
$=(2 \times 3+1)(2 \times 2+1)(2 \times 1+1)(2 \times 1+1)$
$=315$