Question

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

$(1,1)(1,2)-(1,14) \Rightarrow 14$ pts.
If $x=2, y=\frac{27}{2}=13.5$
$(2,2)(2,4) \ldots(2,12) \Rightarrow 6$ 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 $x=6, y=\frac{21}{2}=10.5$
$(6,6) \Rightarrow 1 pt$
If $x=7, y=\frac{39}{4}=9.75$
$(7,7) \Rightarrow 1 pt$
If $x=8, y=9$
$(8,8) \Rightarrow 1 pt$
If $x =9 y =\frac{33}{4}=8.25 \Rightarrow$ no pt.
Total $=31$ pts.