Factorization trees for the number 820, breaking it down into its prime components.

Problem Analysis:

The task involves analyzing two factor trees and determining the prime factorization of the number 820. The goal is to verify the correctness of the factor trees and explain the process step by step.

#### Given Factor Trees:
1. First Factor Tree:
```
820
/ \
410 2
/ \
205 2
/ \
41 5
```

2. Second Factor Tree:
```
820
/ \
205 4
/ \ / \
41 5 2 2
```

Objective:

- Verify the correctness of both factor trees.
- Determine the prime factorization of 820.
- Explain the process of constructing a factor tree and verifying its correctness.

---

Step-by-Step Solution:

#### Step 1: Understanding Factor Trees
A factor tree is a method used to find the prime factorization of a number. The process involves:
1. Starting with the given number (in this case, 820).
2. Breaking it down into two factors.
3. Continuing to break down each non-prime factor until all factors are prime numbers.
4. The leaves of the tree represent the prime factors of the original number.

#### Step 2: Analyzing the First Factor Tree
The first factor tree is:
```
820
/ \
410 2
/ \
205 2
/ \
41 5
```

- Starting with 820:
- Split 820 into 410 and 2. Both 410 and 2 are factors of 820.
- Check if 2 is prime: Yes, 2 is a prime number.
- Move to 410.

- Breaking down 410:
- Split 410 into 205 and 2. Both 205 and 2 are factors of 410.
- Check if 2 is prime: Yes, 2 is a prime number.
- Move to 205.

- Breaking down 205:
- Split 205 into 41 and 5. Both 41 and 5 are factors of 205.
- Check if 41 is prime: Yes, 41 is a prime number.
- Check if 5 is prime: Yes, 5 is a prime number.

- Final Prime Factors:
The leaves of the tree are 41, 5, 2, and 2. Therefore, the prime factorization of 820 is:
$$
820 = 2 \times 2 \times 5 \times 41
$$

#### Step 3: Analyzing the Second Factor Tree
The second factor tree is:
```
820
/ \
205 4
/ \ / \
41 5 2 2
```

- Starting with 820:
- Split 820 into 205 and 4. Both 205 and 4 are factors of 820.
- Check if 4 is composite: Yes, 4 is not a prime number. Break it down further into 2 and 2.
- Move to 205.

- Breaking down 205:
- Split 205 into 41 and 5. Both 41 and 5 are factors of 205.
- Check if 41 is prime: Yes, 41 is a prime number.
- Check if 5 is prime: Yes, 5 is a prime number.

- Final Prime Factors:
The leaves of the tree are 41, 5, 2, and 2. Therefore, the prime factorization of 820 is:
$$
820 = 2 \times 2 \times 5 \times 41
$$

#### Step 4: Verifying Both Factor Trees
Both factor trees lead to the same prime factorization:
$$
820 = 2^2 \times 5 \times 41
$$
This confirms that both factor trees are correct.

---

Final Answer:

The prime factorization of 820 is:
$$
\boxed{2^2 \times 5 \times 41}
$$