To find the prime factorization of 2301 using a factor tree, we will break down the number into its prime factors step by step. Here's how it can be done:

Step 1: Check for divisibility by small primes

First, we'll check for divisibility by small primes like 2, 3, 5, 7, etc.

Since 2301 is odd, it's not divisible by 2. Let's check for 3 by summing the digits of 2301.

$2 + 3 + 0 + 1 = 6$

Since 6 is divisible by 3, 2301 is divisible by 3.

Step 2: Divide 2301 by 3

$2301 \div 3 = 767$

Step 3: Factor 767

Now, we need to factor 767. It’s odd, so it’s not divisible by 2. Let's check divisibility by 3.

Sum of digits of 767 is $7 + 6 + 7 = 20$, which is not divisible by 3. Next, check divisibility by 7.

$767 \div 7 = 109.57$ (not divisible by 7).

Now, check divisibility by 11 using the alternating sum rule: $7 - 6 + 7 = 8$, not divisible by 11. Check divisibility by 13.

$767 \div 13 = 59$.

Step 4: Factor 59

59 is a prime number.

Prime Factorization of 2301

Now, we have fully factored 2301 as:

$2301 = 3 \times 13 \times 59$

Factor Tree Representation

1. Start with 2301 and split it into 3 and 767.
2. Split 767 into 13 and 59.

Here’s the factor tree for 2301:

      2301
     /    \
    3     767
         /   \
       13    59

So, the prime factorization of 2301 is:

$2301 = 3 \times 13 \times 59$

Would you like further details or clarifications? Here are 5 related questions to explore:

1. How can you verify that 59 is a prime number?
2. What is the significance of prime factorization in number theory?
3. How can factor trees help in simplifying fractions?
4. Can prime factorization be used to find the greatest common divisor (GCD) of two numbers?
5. What other methods can you use to find the prime factors of a large number?

Tip: Always start with the smallest primes when creating a factor tree to simplify the process.