The number of positive integral solutions of | vedclass

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(B) Given the equation $a \cdot b \cdot c = 30$,where $a, b, c$ are positive integers.First,find the prime factorization of $30$: $30 = 2^1 \cdot 3^1

Vedclass · Accepted Answer

$27$

Vedclass · Answer

(B) Given the equation $a \cdot b \cdot c = 30$,where $a, b, c$ are positive integers.First,find the prime factorization of $30$: $30 = 2^1 \cdot 3^1 \cdot 5^1$.Each prime factor $(2, 3, 5)$ must be distributed among the three variables $a, b,$ and $c$.For the prime factor $2$,there are $3$ choices (it can go to $a, b,$ or $c$).For the prime factor $3$,there are $3$ choices (it can go to $a, b,$ or $c$).For the prime factor $5$,there are $3$ choices (it can go to $a, b,$ or $c$).Since the distribution of each prime factor is independent,the total number of positive integral solutions is $3 	imes 3 	imes 3 = 27$.

The number of positive integral solutions of $a \cdot b \cdot c = 30$ is

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