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

To solve the problem, we need to find the number of integer points \( P(a, b) \) that lie strictly inside the triangle formed by the x-axis, y-axis, and the line \( 3x + 4y = 60 \). Here, \( a \) is an integer and \( b \) is a multiple of \( a \).

### Step-by-Step Solution:

1. **Find the intercepts of the line**:
- To find the x-intercept, set \( y = 0 \):
\[ 3x + 4(0) = 60 \implies 3x = 60 \implies x = 20 \]
So, the x-intercept is \( (20, 0) \).
- To find the y-intercept, set \( x = 0 \):
\[ 3(0) + 4y = 60 \implies 4y = 60 \implies y = 15 \]
So, the y-intercept is \( (0, 15) \).

2. **Identify the vertices of the triangle**:
- The vertices of the triangle are \( (0, 0) \), \( (20, 0) \), and \( (0, 15) \).

3. **Determine the area of the triangle**:
- The area \( A \) of the triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20 \times 15 = 150 \]

4. **Find the equation of the line in slope-intercept form**:
- Rearranging the line equation \( 3x + 4y = 60 \) gives:
\[ 4y = 60 - 3x \implies y = 15 - \frac{3}{4}x \]

5. **Identify integer points \( P(a, b) \)**:
- We need to find integer points \( (a, b) \) such that \( b \) is a multiple of \( a \) and lies strictly inside the triangle.
- The points must satisfy the line equation:
\[ b < 15 - \frac{3}{4}a \]

6. **Count the valid points for each integer \( a \)**:
- For \( a = 1 \):
\[ b < 15 - \frac{3}{4}(1) = 15 - 0.75 = 14.25 \implies b = 1, 2, \ldots, 14 \quad \text{(14 values)} \]
- For \( a = 2 \):
\[ b < 15 - \frac{3}{4}(2) = 15 - 1.5 = 13.5 \implies b = 2, 4, 6, 8, 10, 12, 14 \quad \text{(6 values)} \]
- For \( a = 3 \):
\[ b < 15 - \frac{3}{4}(3) = 15 - 2.25 = 12.75 \implies b = 3, 6, 9, 12 \quad \text{(4 values)} \]
- For \( a = 4 \):
\[ b < 15 - \frac{3}{4}(4) = 15 - 3 = 12 \implies b = 4, 8, 12 \quad \text{(3 values)} \]
- For \( a = 5 \):
\[ b < 15 - \frac{3}{4}(5) = 15 - 3.75 = 11.25 \implies b = 5, 10 \quad \text{(2 values)} \]
- For \( a = 6 \):
\[ b < 15 - \frac{3}{4}(6) = 15 - 4.5 = 10.5 \implies b = 6 \quad \text{(1 value)} \]
- For \( a = 7 \):
\[ b < 15 - \frac{3}{4}(7) = 15 - 5.25 = 9.75 \implies \text{No valid } b \]

7. **Total the valid points**:
- Total points = \( 14 + 6 + 4 + 3 + 2 + 1 = 30 \).

### Final Answer:
The number of points \( P(a, b) \) that lie strictly inside the triangle is **30**.