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A triangle is formed by $X -$ axis, $Y$ - axis and the line $3 x+4 y=60$. 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 $...........$
$31$
$30$
$28$
$56$
Solution
If $x=1, y=\frac{57}{4}=14.25$
$(1,1)(1,2)-(1,14) \quad \Rightarrow 14$ pts.
If $x =2, y =\frac{27}{2}=13.5$
$(2,2)(2,4) \ldots(2,12) \quad \Rightarrow 6$ pts.
If $x=3, y=\frac{51}{4}=12.75$
$(3,3)(3,6)-(3,12) \quad \Rightarrow 4$ pts.
If $x=4, y=12$
$(4,4)(4,8) \quad \Rightarrow 2$ pts.
If $x=5 . y=\frac{45}{4}=11.25$
$(5,5),(5,10) \quad \Rightarrow 2$ pts.
If $x=6, y=\frac{21}{2}=10.5$
$(6,6) \quad \Rightarrow 1 pt$.
If $x=7, y=\frac{39}{4}=9.75$
$(7,7) \quad \Rightarrow 1 pt$.
If $x=8, y=9$
$(8,8) \quad \Rightarrow 1 pt$.
If $x =9 y =\frac{33}{4}=8.25 \Rightarrow$ no pt.
Total $=31$ pts.
