Question

Find the number of positive integral solutions of $abc=45$

Answer 1

$abc=45=3^2\times 5$

$\Rightarrow a=3^{a_1}{5}^{a_2};b=3^b,5^{b_2},c=3^a,5^{c_2}$

$\Rightarrow 3^{a_1+b_1+c_1}{5}^{a_2+b_2+c_2}=3^2\times 5$

$\Rightarrow a_1+b_1+c_1=2;a_2+b{n}_2+c_2=1$

we know that

$a_1+a_2+.....a_r=\ne$ has $\ne {+}^{r-1}{C}_{r-1}$ integer solution

$a_1+b_1+c_1=2$ has $2{+}^{3-1}{C}_{3-1}{=}^4{C}_2=6$ solution in integer

$a_2+b_2+c_2=1$ has ${}^{1+3-1}{C}_{3-1}{=}^3{C}_2=3$ solution in integers

So, required no. of positive integer solution

$=6\times 3=18$