A triangle is formed by X-axis, Y-axis and the line $$3x + 4y = 60$$. Then the number of points $$P(a, b)$$ which lie strictly inside the triangle, where $$a$$ is an integer and $$b$$ is a multiple of $$a$$, is _____.

A triangle is formed by the X-axis, Y-axis, and the line $$3x + 4y = 60$$. We need to find the number of points $$P(a, b)$$ strictly inside the triangle where $$a$$ is a positive integer and $$b$$ is a positive multiple of $$a$$.

Identify the triangle.

The vertices of the triangle are:

Origin: $$(0, 0)$$

X-intercept: $$(20, 0)$$

Y-intercept: $$(0, 15)$$

Set up the conditions.

For a point $$(a, b)$$ to be strictly inside the triangle:

$$a > 0$$, $$b > 0$$, and $$3a + 4b < 60$$

Also, $$a$$ is a positive integer and $$b = ka$$ for some positive integer $$k$$.

Count the valid points for each value of $$a$$.

For each $$a$$, we need $$3a + 4ka < 60$$, i.e., $$a(3 + 4k) < 60$$, i.e., $$k < \frac{60/a - 3}{4}$$.

$$a = 1$$: $$k < \frac{57}{4} = 14.25$$ → $$k = 1, 2, \ldots, 14$$ → 14 points

$$a = 2$$: $$k < \frac{27}{4} = 6.75$$ → $$k = 1, 2, \ldots, 6$$ → 6 points

$$a = 3$$: $$k < \frac{17}{4} = 4.25$$ → $$k = 1, 2, 3, 4$$ → 4 points

$$a = 4$$: $$k < \frac{12}{4} = 3$$ → $$k = 1, 2$$ → 2 points

$$a = 5$$: $$k < \frac{9}{4} = 2.25$$ → $$k = 1, 2$$ → 2 points

$$a = 6$$: $$k < \frac{7}{4} = 1.75$$ → $$k = 1$$ → 1 point

$$a = 7$$: $$k < \frac{39/7}{4} = \frac{39}{28} = 1.393$$ → $$k = 1$$ → 1 point

$$a = 8$$: $$k < \frac{36/8}{4} = \frac{36}{32} = 1.125$$ → $$k = 1$$ → 1 point

$$a = 9$$: $$k < \frac{33/9}{4} = \frac{33}{36} = 0.917$$ → no valid $$k$$ → 0 points

For $$a \geq 9$$: $$\frac{60/a - 3}{4} \leq 1$$, so no valid positive integer $$k$$ exists.

Total count.