Find the prime factorization of 1 and 20 by both the factor tree method and the continuous division method.

Find the prime factorization of 1 and 20 by both the factor tree method and the continuous division method. Try to create two factor trees.

The Correct Answer and Explanation is:

Let’s begin with the prime factorization of 1 and 20 using both the factor tree method and the continuous division method.

Prime Factorization of 1

• Factor Tree Method:
  The prime factorization of 1 is a unique case because 1 is not divisible by any prime number. As a result, 1 is considered neither prime nor composite. It does not have a prime factorization, so there is no factor tree for 1.
• Continuous Division Method:
  Similarly, using continuous division, we cannot divide 1 by any prime number. Thus, 1 doesn’t have a prime factorization.

Prime Factorization of 20

Now, let’s focus on 20, and we’ll explore both methods.

Factor Tree Method:

1. Start with 20.
2. 20 is divisible by 2 (which is a prime number), so we split it:
   20÷2=1020 \div 2 = 1020÷2=10, so we now have:
   20=2×1020 = 2 \times 1020=2×10
3. Next, break 10 into prime factors. 10 is divisible by 2, so:
   10÷2=510 \div 2 = 510÷2=5, giving us:
   10=2×510 = 2 \times 510=2×5
4. Now, 5 is a prime number, so we stop here.

So, the factor tree for 20 is:

    20
     /  \
    2   10
        /  \
       2    5

This gives the prime factorization:20=2×2×5=22×520 = 2 \times 2 \times 5 = 2^2 \times 520=2×2×5=22×5

Continuous Division Method:

1. Start with 20 and divide by the smallest prime number, 2:
   20÷2=1020 \div 2 = 1020÷2=10
2. Divide 10 by 2 again:
   10÷2=510 \div 2 = 510÷2=5
3. Now, divide 5 by 5 (since it is a prime number):
   5÷5=15 \div 5 = 15÷5=1

This results in the prime factorization:20=2×2×5=22×520 = 2 \times 2 \times 5 = 2^2 \times 520=2×2×5=22×5

Summary:

For 1, there is no prime factorization, and no factor tree can be created.

For 20, the prime factorization is:20=22×520 = 2^2 \times 520=22×5

This was confirmed using both the factor tree method and the continuous division method.

The continuous division method involves dividing by prime numbers until you reach 1, while the factor tree method visually breaks down the number into its prime factors. Both methods give us the same result for 20.