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Let's solve the problem step-by-step.

Given:
- The length \( l \) and width \( w \) of a rectangle are prime numbers.
- The perimeter of the rectangle is 36 meters.

We need to find the largest possible area of the rectangle.

### Step 1: Formulate the Perimeter Equation

The perimeter \( P \) of a rectangle is given by:
\[ P = 2l + 2w \]
Given \( P = 36 \):
\[ 2l + 2w = 36 \]
\[ l + w = 18 \]

### Step 2: Identify Prime Number Pairs

We need to find pairs of prime numbers whose sum is 18. The prime numbers less than 18 are:
\[ 2, 3, 5, 7, 11, 13, 17 \]

Checking pairs that sum to 18:
- \( 5 + 13 = 18 \)
- \( 7 + 11 = 18 \)

### Step 3: Calculate the Areas

Calculate the area for each pair:
1. For \( l = 5 \) and \( w = 13 \):
   \[ \text{Area} = l \times w = 5 \times 13 = 65 \, \text{square meters} \]

2. For \( l = 7 \) and \( w = 11 \):
   \[ \text{Area} = l \times w = 7 \times 11 = 77 \, \text{square meters} \]

### Step 4: Determine the Largest Area

Compare the areas calculated:
- \( 65 \, \text{square meters} \) (from \( l = 5 \) and \( w = 13 \))
- \( 77 \, \text{square meters} \) (from \( l = 7 \) and \( w = 11 \))

The largest area is \( 77 \, \text{square meters} \).

### Conclusion

The largest number of square meters the area could be is:
\[ \boxed{77} \]

Would you like further details or have any questions? 

Here are five related questions to expand on this topic:
1. What if the perimeter was a different value, like 40 meters? How would you find the largest area?
2. How do we handle a scenario where there are no pairs of prime numbers that sum to a given perimeter?
3. How can we generalize the solution for any given perimeter \( P \)?
4. What are the properties of prime numbers that make them useful in problems like this?
5. How would the solution change if one of the dimensions was restricted to non-prime numbers?

**Tip:** When solving problems involving prime numbers and specific conditions, always list all relevant prime numbers within the range and verify their sums or products to meet the given criteria.

Suppose the number of units of the length and width of a rectangle are prime numbers and the perimeter is 36 meters. What is the largest number of square meters the area could be?

Learn how to solve a problem where the perimeter of a rectangle is 36 meters, and both length and width are prime numbers. Using algebra and geometry concepts, we calculate the largest possible area of the rectangle, which is 77 square meters.

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