Residents were surveyed in order to determine which flowers to plant in the new Public Garden. A total of $N$ people participated in the survey. Exactly $\frac{9}{14}$ of those surveyed said that the colour of the flower was important. Exactly $\frac{7}{12}$ of those surveyed said that the smell of the flower was important. In total, 753 people said that both the colour and smell were important. How many possible values are there for $N$ ?
(A) 22
(B) 23
(C) 21
(D) 24
(E) 25

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-042.jpg?height=195&width=198&top_left_y=233&top_left_x=389)
A total of $N$ people participated in the survey.

Exactly $\frac{9}{14}$ of those surveyed, or $\frac{9}{14} \times N$ people, said that the colour of the flower was important. Exactly $\frac{7}{12}$ of those surveyed, or $\frac{7}{12} \times N$ people, said that the smell of the flower was important. Since the number of people surveyed must be a whole number, then $N$ must be a multiple of 14 and $N$ must also be a multiple of 12 .

The lowest common multiple of 14 and 12 is $2 \times 7 \times 6$ or 84 , and so $N$ is a multiple of 84 .

Since $N$ is a multiple of 84 , let $N=84 k$ where $k$ is some positive integer.

That is, $\frac{9}{14} \times N=\frac{9}{14}(84 k)=54 k$ people said that the colour of the flower was important and $\frac{7}{12} \times N=\frac{7}{12}(84 k)=49 k$ people said that the smell of the flower was important. The information can be represented with a Venn diagram, as shown below.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-201.jpg?height=428&width=656&top_left_y=1239&top_left_x=816)

Since $N=84 k$, we must find both a maximum and a minimum value for $k$ in order to determine the number of possible values for $N$.

Finding a Minimum Value for $k$

We know that 753 people said that both the colour and the smell were important.

We also know that $54 k$ said that the colour was important and that $49 k$ said that the smell was important.

The 753 people who said that both were important are included among both the $54 k$ people and among the $49 k$ people.

Therefore, $54 k$ must be at least 753 and $49 k$ must also be at least 753 .

Since $54 k$ is larger than $49 k$, then we only need to find $k$ for which $49 k$ is larger than or equal to 753 .

We note that $49 \times 15=735$ and that $49 \times 16=784$.

Therefore, the smallest value of $k$ for which $49 k$ is at least 753 is $k=16$.

As noted above, $k=16$ will also make $54 k$ larger than 753 .

Therefore, the minimum possible value for $k$ is 16 .

## Finding a Maximum Value for $k$

There were $84 k$ people surveyed of whom $54 k$ said that the colour was important.

This means that there are $84 k-54 k=30 k$ people who said that the colour was not important. We note that $49 k$ people said that the smell was important. These $49 k$ people include some or all of the $30 k$ people who said that the colour was not important.

In other words, at most $30 k$ of the $49 k$ people said that the smell was important and the colour was not important, which means that at least $49 k-30 k=19 k$ people said that the smell was important and that the colour was important.

Since we know that 753 people said that both the smell and colour were important, then 753 is at least as large as $19 k$.

Since $19 \times 40=760$ and $19 \times 39=741$, then $k$ must be at most 39 .

The minimum value of $k$ is 16 and the maximum value of $k$ is 39 .

Thus, there are $39-16+1=24$ possible values for $k$ and since $N=84 k$, there are also 24 possible values for $N$.

Note: We should also justify that each value of $k$ between 16 and 39 is possible.

Using the Venn diagram above, we see that the total of $84 k$ people surveyed means that $84 k-(54 k-753)-753-(49 k-753)=753-19 k$ people fall outside the two circles.

From our work in finding the minimum above, we can see that for each $k$ between 16 and 39 , inclusive, each of the three integers $54 k-753,753$, and $49 k-753$ is positive, since $k$ is at least 16. Also, from our work in finding the maximum above, we can see that for each $k$ between 16 and 39 , inclusive, the integer $753-19 k$ is positive, since $k$ is at most 39 .

Therefore, all four quantities are positive integers for each of these values of $k$, which means that we can construct a Venn diagram with these numbers, as required.

ANsweR: (D)