How many of the positive divisors of 3240 are multiples of 3?
\(\text {The prime factors of $\mathbf{3240= 2^{\color{red}3} \cdot 3^{\color{red}4} \cdot 5^{\color{red}1} }$ } \\ \text {The positive divisors of $3240$ are $\mathbf{({\color{red}3}+1)({\color{red}4}+1)({\color{red}1}+1) = 40 }$ } \)
\(\text{How many of the positive divisors of $3240$ are $\textbf{not}$ multiples of $3$ ?}\)
\(\begin{array}{|ccccccl|} \hline 2^3 && 3^4 && 5^1 \\ \begin{Bmatrix} 2^0 \\ 2^1 \\ 2^2 \\ 2^3 \end{Bmatrix} && \begin{Bmatrix} 3^0 \end{Bmatrix} && \begin{Bmatrix} 5^0 \\ 5^1 \end{Bmatrix} \\ 4 & \times& 1 & \times& 2 &=& \mathbf{8} \\ \hline \end{array}\)
\(\text{How many of the positive divisors of $3240$ are multiples of $3$ ?}\)
\(\mathbf{40-8 = 32} \)
\(\text{There are $\mathbf{32}$ positive divisors of $3240$, which are multiples of $3$.}\)
