Question

The number of positive integral solutions of the equation $x_1{x}_2{x}_3{x}_4{x}_5=1050$ is $375n$ when $n\in N$. Then $n=$

Answer 1

$x_1{x}_2{x}_3{x}_4{x}_5=2\times 3\times 5^2\times 7$
we can assign $2,3$ or $7$ to any of $5$ variables in $5\times 5\times 5$.
We can assign $5^2$ to just one variable in $5$ ways or we can assign $5^2=5\times 5$ to two variables in ${}^5{C}_2$ ways.
$\therefore$ $5$ can be assigned in ${}^5{C}_1{+}^5{C}_2=5+10=15$ ways.
Hence required number of solutions=$5\times 5\times 5\times 15=1875$