A triangle is formed by X-axis, Y-axis and the line 3x+4y=60.

Question

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 \_\_\_\_\_\_\_\_.

Answer: 31

Accepted Answer

For each integer $a=1,\dots,9$ (since $x<20$ within the triangle roughly), the line gives $y<\dfrac{60-3a}4$, and we count multiples of $a$ strictly between $0$ and that bound.

- $a=1$: $y<14.25$, multiples of 1: $1$ to $14$ \to  14 points
- $a=2$: $y<13.5$, multiples of 2: $2,4,\dots,12$ \to  6 points
- $a=3$: $y<12.75$, multiples of 3: $3,6,9,12$ \to  4 points
- $a=4$: $y<12$, multiples of 4: $4,8$ \to  2 points
- $a=5$: $y<11.25$, multiples of 5: $5,10$ \to  2 points
- $a=6$: $y<10.5$, multiples of 6: $6$ \to  1 point
- $a=7$: $y<9.75$, multiples of 7: $7$ \to  1 point
- $a=8$: $y<9$, multiples of 8: $8$ \to  1 point
- $a=9$: $y<8.25$, no multiple of 9 fits

$$\text{Total} = 14+6+4+2+2+1+1+1 = 31$$

Solutions · 1