Question

als garden is 18 feet long and 30 feet wide. he wants to put fence posts the same distance apart along all sides of the garden. what is the greatest distance apart that al can place the fence posts?

Explanation:

Step1: Identify the problem type

This is a problem about finding the greatest common divisor (GCD) of the length and width of a rectangular garden to determine the maximum distance between fence posts. The garden has dimensions 18 feet (length) and 30 feet (width).

Step2: Recall the GCD formula

To find the GCD of two numbers, we can use the Euclidean algorithm or factorize the numbers. Let's factorize 18 and 30.

• Prime factorization of 18: \(18 = 2 \times 3^2\)
• Prime factorization of 30: \(30 = 2 \times 3 \times 5\)

Step3: Determine the common factors

The common prime factors of 18 and 30 are 2 and 3.

Step4: Calculate the GCD

To find the GCD, we take the product of the lowest powers of the common prime factors. For 2, the lowest power is \(2^1\), and for 3, the lowest power is \(3^1\). So, \(GCD(18, 30)=2\times3 = 6\).

Answer:

The greatest distance apart that Al can place the fence posts is 6 feet.