1. Find the triangle region

The line is $3x+4y=60$ with intercepts:

• on the $x$-axis: $y=0 \Rightarrow x=20$
• on the $y$-axis: $x=0 \Rightarrow y=15$

So the triangle is bounded by: $x=0,\quad y=0,\quad 3x+4y=60$

Points strictly inside satisfy $x>0,\quad y>0,\quad 3x+4y<60$

We want points $P(a,b)$ such that:

• $a$ is an integer
• $b$ is a multiple of $a$, i.e. $b=ka$ for some positive integer $k$
• and $(a,b)$ lies strictly inside the triangle.

2. Use the condition $b$ is a multiple of $a$

Let $b=ka,$ where $k\in \mathbb{Z}_{>0}$ since $b>0$ inside the triangle.

Substitute into the inequality: $3a+4b<60$ $3a+4ka<60$ $a(3+4k)<60$

For each positive integer $k$, the number of positive integers $a$ satisfying this is: $a<\frac{60}{3+4k}$

3. Count possible values of $a$ for each $k$

We need positive integers $a$ such that $a<\frac{60}{3+4k}.$ Also $k$ must be such that at least $a=1$ is possible: $3+4k<60 \Rightarrow 4k<57 \Rightarrow k\le 14.$

Now count for $k=1$ to $14$:

$k=1$

$a<\frac{60}{7}=8.57\ldots$ So $a=1,2,3,4,5,6,7,8$ : $8$ values

$k=2$

$a<\frac{60}{11}=5.45\ldots$ So $a=1,2,3,4,5$ : $5$ values

$k=3$

$a<\frac{60}{15}=4$ Strictly less, so $a=1,2,3$ : $3$ values

$k=4$

$a<\frac{60}{19}=3.15\ldots$ So $a=1,2,3$ : $3$ values

$k=5$

$a<\frac{60}{23}=2.60\ldots$ So $a=1,2$ : $2$ values

$k=6$

$a<\frac{60}{27}=2.22\ldots$ So $a=1,2$ : $2$ values

$k=7$

$a<\frac{60}{31}=1.93\ldots$ So $a=1$ : $1$ value

$k=8$

$a<\frac{60}{35}=1.71\ldots$ So $a=1$ : $1$ value

$k=9$

$a<\frac{60}{39}=1.53\ldots$ So $a=1$ : $1$ value

$k=10$

$a<\frac{60}{43}=1.39\ldots$ So $a=1$ : $1$ value

$k=11$

$a<\frac{60}{47}=1.27\ldots$ So $a=1$ : $1$ value

$k=12$

$a<\frac{60}{51}=1.17\ldots$ So $a=1$ : $1$ value

$k=13$

$a<\frac{60}{55}=1.09\ldots$ So $a=1$ : $1$ value

$k=14$

$a<\frac{60}{59}=1.01\ldots$ So $a=1$ : $1$ value

4. Add all counts

Total number of points: $8+5+3+3+2+2+1+1+1+1+1+1+1+1$

First sum larger terms: $8+5+3+3+2+2=23$ Remaining eight $1$'s give: $23+8=31$

Thus the required number of points is $\boxed{31}$

5. Comparison with stored answer

Stored correct answer = $31$

Our derived answer is also $31$, so it agrees.