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 ____

Hint:

When solving geometry problems with constraints, carefully analyze integer solutions and relationships between coordinates

Answer 1

Correct Answer: 31

The intercepts of the line are (20,0) and (0,15). Compute the points inside the triangle satisfying b = ka. The total is: 31points.

Answer 2

The correct answer is 31

\((1,1)(1,2)−(1,14)⇒14 pts.\)

If \(x=2\), \(y=\frac{27}{2} = 13.5\)

\((2,2)(2,4)…(2,12)⇒6 pts.\)

If \(x=3\), \(y=\frac{51}{4} = 12.75\)

\((3,3)(3,6)−(3,12)⇒4 pts.\)

If \(x=4\), \(y=12\)

\((4,4)(4,8)⇒2 pts.\)

If \(x=5\), \(y=\frac{45}{4} = 11.25\)

\((5,5),(5,10)⇒2 pts.\)

If \(x=6\), \(y=\frac{21}{2} = 10.5\)

\((6,6)⇒1pt\)

If \(x=7\), \(y=\frac{39}{4} = 9.75\)

(7,7)⇒1pt

If \(x=8\), \(y=9\)

\((8,8)⇒1pt\)

If \(x=9\) \(y=\frac{33}{4}= 8.25\)\(\Rightarrow\) no pt.

Total = 31 pts.