Question

The number of positive integral solutions of $a·b·c=30$ is$?$

• A. $30$
• B. $27$
• C. $8$
• D. $6$

Answer 1

The correct option is B

$27$

Find the number of positive integral solutions of the given equation.

An equation $a·b·c=30$ is given.

Since the positive factors of $30$ are $1,2,3,5,6,10,15,30$.

Case $1$: $1·3·10=30$.

So, the number of integral solutions for this case is $3!$.

Case $2$: $1·2·15=30$.

So, the number of integral solutions for this case is $3!$.

Case $3$: $1·1·30=30$.

So, the number of integral solutions for this case is $\frac{3!}{2!}$.

Case $4$: $1·5·6=30$.

So, the number of integral solutions for this case is $3!$.

Case $5$: $2·3·5=30$.

So, the number of integral solutions for this case is $3!$.

Thus, the total number of integral solutions $4·3!+\frac{3!}{2!}=27$.

Therefore, the number of positive integral solutions of $a·b·c=30$ is $27$.

Hence, option (B) is the correct answer.