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 start by finding the prime factorization of 1 and 20 using both methods:
1. Prime Factorization of 1
- The number 1 is unique because it is neither a prime number nor a composite number. It does not have any prime factors, as a prime factorization requires at least one prime number, and 1 is not divisible by any prime numbers.
- Prime Factorization of 1:
There is no prime factorization for 1 because it doesn’t have any prime divisors.
2. Prime Factorization of 20
Now let’s find the prime factorization of 20. We will use both methods for this.
Method 1: Factor Tree Method
To build a factor tree for 20:
- Start with 20 and break it down into two factors: 20=2×1020 = 2 \times 1020=2×10.
- Now, factor 10: 10=2×510 = 2 \times 510=2×5.
This gives us the prime factorization:20=2×2×520 = 2 \times 2 \times 520=2×2×5
or, written as:20=22×520 = 2^2 \times 520=22×5
Factor Tree for 20:
20
/ \
2 10
/ \
2 5
Method 2: Continuous Division Method
To factor 20 using continuous division:
- Start by dividing 20 by the smallest prime number, which is 2: 20÷2=1020 \div 2 = 1020÷2=10.
- Now, divide 10 by 2 again: 10÷2=510 \div 2 = 510÷2=5.
- Finally, 5 is already a prime number, so we stop here.
The prime factorization of 20 is:20=2×2×520 = 2 \times 2 \times 520=2×2×5
or:20=22×520 = 2^2 \times 520=22×5
Conclusion:
Both methods give us the same prime factorization for 20:20=22×520 = 2^2 \times 520=22×5
This shows that 20 is made up of two 2’s and one 5, and there are no other prime factors.
