Key Concepts and Formulas

• The equation of a line $ax + by = c$ defines a region in the Cartesian plane. Points strictly inside the triangle formed by the axes and the line satisfy $x > 0$, $y > 0$, and $ax + by < c$.
• Integer solutions can be found by systematically checking integer values for one variable and solving for the other.
• The number of multiples of 'a' less than a given number 'y' can be found by $\lfloor \frac{y}{a} \rfloor$, where $\lfloor x \rfloor$ is the floor function representing the greatest integer less than or equal to x.

Step-by-Step Solution

Step 1: Define the region of interest. The triangle is formed by the x-axis ($y=0$), the y-axis ($x=0$), and the line $3x + 4y = 60$. Since we are looking for points strictly inside the triangle, we need $x > 0$, $y > 0$, and $3x + 4y < 60$.

Step 2: Express $y$ in terms of $x$. We can rewrite the inequality $3x + 4y < 60$ as $4y < 60 - 3x$, which gives us $y < \frac{60 - 3x}{4}$.

Step 3: Determine the possible integer values of $x$. Since $x > 0$ and $y > 0$, we must have $\frac{60 - 3x}{4} > 0$, which implies $60 - 3x > 0$, so $3x < 60$, and $x < 20$. Therefore, $x$ can take integer values from 1 to 19.

Step 4: Iterate through possible values of $x$ and count the valid points. We are given that $y = ab$, where $b$ is an integer. Since $y > 0$, we need to count the number of multiples of $a$, where $a$ is the x-coordinate, that are strictly less than $\frac{60 - 3x}{4}$. In our case, $a = x$. So, we need to find the number of integer multiples of $x$ that are less than $\frac{60 - 3x}{4}$. Thus, we want to find the number of integers $b$ such that $xb < \frac{60 - 3x}{4}$. This is equivalent to $b < \frac{60 - 3x}{4x}$. The number of such integers is $\lfloor \frac{60 - 3x}{4x} \rfloor$.

Let's analyze this for $x=1, 2, 3, ...$:

• If $x = 1$, then $y < \frac{60 - 3}{4} = \frac{57}{4} = 14.25$. Since $y$ is a multiple of $x=1$, $y$ can be $1, 2, 3, ..., 14$. The number of such points is $\lfloor \frac{57}{4} \rfloor = 14$.

• If $x = 2$, then $y < \frac{60 - 6}{4} = \frac{54}{4} = 13.5$. Since $y$ is a multiple of $x=2$, $y$ can be $2, 4, 6, ..., 12$. The number of such points is $\lfloor \frac{54}{8} \rfloor = \lfloor \frac{27}{4} \rfloor = 6$.

• If $x = 3$, then $y < \frac{60 - 9}{4} = \frac{51}{4} = 12.75$. Since $y$ is a multiple of $x=3$, $y$ can be $3, 6, 9, 12$. The number of such points is $\lfloor \frac{51}{12} \rfloor = \lfloor \frac{17}{4} \rfloor = 4$.

• If $x = 4$, then $y < \frac{60 - 12}{4} = \frac{48}{4} = 12$. Since $y$ is a multiple of $x=4$, $y$ can be $4, 8$. The number of such points is $\lfloor \frac{48}{16} \rfloor = 2$.

• If $x = 5$, then $y < \frac{60 - 15}{4} = \frac{45}{4} = 11.25$. Since $y$ is a multiple of $x=5$, $y$ can be $5, 10$. The number of such points is $\lfloor \frac{45}{20} \rfloor = \lfloor \frac{9}{4} \rfloor = 2$.

• If $x = 6$, then $y < \frac{60 - 18}{4} = \frac{42}{4} = 10.5$. Since $y$ is a multiple of $x=6$, $y$ can be $6$. The number of such points is $\lfloor \frac{42}{24} \rfloor = \lfloor \frac{7}{4} \rfloor = 1$.

• If $x = 7$, then $y < \frac{60 - 21}{4} = \frac{39}{4} = 9.75$. Since $y$ is a multiple of $x=7$, $y$ can be $7$. The number of such points is $\lfloor \frac{39}{28} \rfloor = 1$.

• If $x = 8$, then $y < \frac{60 - 24}{4} = \frac{36}{4} = 9$. Since $y$ is a multiple of $x=8$, $y$ can be $8$. The number of such points is $\lfloor \frac{36}{32} \rfloor = 1$.

• If $x = 9$, then $y < \frac{60 - 27}{4} = \frac{33}{4} = 8.25$. Since $y$ is a multiple of $x=9$, we have $9b < 8.25$. Since $b \ge 1$, no solutions exist.

• If $x = 10$, then $y < \frac{60 - 30}{4} = \frac{30}{4} = 7.5$. Since $y$ is a multiple of $x=10$, we have $10b < 7.5$. Since $b \ge 1$, no solutions exist.

We can observe that for $x \ge 9$, $\frac{60 - 3x}{4x} < 1$, so there are no solutions.

Step 5: Calculate the total number of points. The total number of points is $14 + 6 + 4 + 2 + 2 + 1 + 1 + 1 = 31$.

Common Mistakes & Tips

• Be careful when calculating the upper bound for $y$. Ensure you are using the strict inequality.
• Remember that $x$ and $y$ must be strictly positive.
• Don't forget to use the floor function to find the number of integers.

Summary

We found the region defined by the triangle and the condition that the points must lie strictly inside. We then iterated through possible integer values of $x$ and calculated the number of multiples of $x$ that lie within the allowed range for $y$. Finally, we summed the number of points for each $x$ value to find the total number of such points.

The final answer is $\boxed{31}$.