Question

Number of positive integral solutions of $abc=30$ is

• A. $9$
• B. $27$
• C. $81$
• D. $243$

Answer 1

The correct option is B $27$

Prime factorization of $30=2\times 3\times 5$
abc = $2\times 3\times 5$. Possible ordered pair of (a,b,c) are $(1,1,30),(1,2,15),(1,3,10),(1,5,6),(2,3,5)$

Number of permutations of each ordered pair is
$\frac{3!}{2!}\text{for}(1,1,30)$
$3!\text{for}(1,2,15)$
$3!\text{for}(1,3,10)$
$3!\text{for}(1,5,6)$
$3!\text{for}(2,3,5)$

There are $=3+6+6+6+6=27$ possible integral solutions of $abc=27$