How many of the positive divisors of 3240 are multiples of 3?

It's 32 and especially this :

3  6 9 12 15 18 24 27 30 36  45 54 60 72 81 90 108 120 135 162 180 216 270 324 360 405 540 648 810 1080 1620 3240

Solution _{^{(using a mathematical construction)}}:

\(\text {The prime factors of } 3240= 2^3 \cdot 3^4 \cdot 5^1 \\ \text {Assemble the primes as multiples of three (3), and count each construction. } \\ \text {For 3240: }\\ \hspace{.8cm} \text {There can be (0, 1, 2, or 3) twos (2) in the construction of a divisor. }\\ \hspace{1.cm} \small \text { So there are a total of four (4) ways to use a two (2) –one of the ways is to} \textbf { not} \text { use it. } \\ \small \text { }\\ \hspace{.8cm} \text {There can be (1, 2, 3 or 4) threes (3) in the construction of a divisor. }\\ \hspace{1.cm} \small \text {Note that for a number to be a multiple of three (3), the number must have at least one multiple of three (3) - }\\ \hspace{1 cm} \small \text {that is, there cannot be a zero (0) number of threes (3). }\\ \hspace{1.cm} \small \text {So there are a total of four (4) ways to use a three (3). }\\ \text { }\\ \hspace{.8cm} \text {There can be (0 or 1) fives (5) in the construction of a divisor. }\\ \hspace{1.cm} \small \text { So there are a total of two (2) ways to use a five (5) –one of the ways is to not use it.}\\ \text { }\\ \text {The product of these counts is } 4\cdot 4\cdot 2 = 32 \\ \text { }\\ \text {There are } \textbf {32 divisors of 3240 that are multiples of three (3). }\\ \)

GA

How many of the positive divisors of 3240 are multiples of 3?

\(\text {The prime factors of $\mathbf{3240= 2^{\color{red}3} \cdot 3^{\color{red}4} \cdot 5^{\color{red}1} }$ } \\ \text {The positive divisors of $3240$ are $\mathbf{({\color{red}3}+1)({\color{red}4}+1)({\color{red}1}+1) = 40 }$ } \)

\(\text{How many of the positive divisors of $3240$ are $\textbf{not}$ multiples of $3$ ?}\)

\(\begin{array}{|ccccccl|} \hline 2^3 && 3^4 && 5^1 \\ \begin{Bmatrix} 2^0 \\ 2^1 \\ 2^2 \\ 2^3 \end{Bmatrix} && \begin{Bmatrix} 3^0 \end{Bmatrix} && \begin{Bmatrix} 5^0 \\ 5^1 \end{Bmatrix} \\ 4 & \times& 1 & \times& 2 &=& \mathbf{8} \\ \hline \end{array}\)

\(\text{How many of the positive divisors of $3240$ are multiples of $3$ ?}\)

\(\mathbf{40-8 = 32} \)

\(\text{There are $\mathbf{32}$ positive divisors of $3240$, which are multiples of $3$.}\)