1. **State the problem:** We need to find the equation of a water pipe line whose gradient is the reciprocal of a given prime number. The line must pass through the point $P(a,b)$, where $a$ is the highest common factor (HCF) of 18 and 24, and $b$ is the smallest square number greater than 5. 2. **Find $a$ (HCF of 18 and 24):** - Factors of 18: 1, 2, 3, 6, 9, 18 - Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Common factors: 1, 2, 3, 6 - Highest common factor: $a = 6$ 3. **Find $b$ (smallest square number greater than 5):** - Square numbers: 1, 4, 9, 16, ... - The smallest square number greater than 5 is $b = 9$ 4. **Define the gradient:** - Let the given prime number be $p$. - The gradient $m$ is the reciprocal of $p$, so $m = \frac{1}{p}$. 5. **Equation of the line:** - The general equation of a line with gradient $m$ passing through point $(a,b)$ is: $$y - b = m(x - a)$$ - Substitute $a=6$, $b=9$, and $m=\frac{1}{p}$: $$y - 9 = \frac{1}{p}(x - 6)$$ 6. **Final equation:** $$y = \frac{1}{p}(x - 6) + 9$$ This is the equation of the main water pipe line with gradient reciprocal of a prime number $p$ passing through point $(6,9)$.