Question

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

Answer 1

Answer: 31

$(1,1)(1,2)-(1,14)\Rightarrow 14$ pts.
If $x=2,y=\frac{27}{2}=13.5$
$(2,2)(2,4)\dots (2,12)\Rightarrow 6$ pts.
If $x=3,y=\frac{51}{4}=12.75$
$(3,3)(3,6)-(3,12)\Rightarrow 4$ pts.
If $x=4,y=12$
$(4,4)(4,8)\Rightarrow 2$ pts.
If $x=5\cdot y=\frac{45}{4}=11.25$
$(5,5),(5,10)\Rightarrow 2$ pts.
If $x=6,y=\frac{21}{2}=10.5$
$(6,6)\Rightarrow 1pt$
If $x=7,y=\frac{39}{4}=9.75$
$(7,7)\Rightarrow 1pt$
If $x=8,y=9$
$(8,8)\Rightarrow 1pt$
If $x=9y=\frac{33}{4}=8.25\Rightarrow$ no pt.
Total $=31$ pts.