Question

Using the Factor Tree Method, find the prime factorization of the following:
(a) 8
(b) 20
(c) 34
(d) 44
(e) 21
(f) 42
(g) 38
(h) 45

Answer 1

Hint: We start solving the problem by writing the given number as a multiplication of two numbers out of which one will be a prime number. We then write the other number as a multiplication of two numbers out of which one will be a prime number. We continue this process till we get the number as a product of all prime numbers. Similarly, we follow this for all the given numbers to get the required result.

Complete step-by-step answer:
According to the problem, we are given that we need to find the prime factorization of the given numbers using the factor tree method.
(a) We need to find the prime factorization of the 8.
We know that $8=2\times 4$.
\[\Rightarrow 8=2\times 2\times 2\]. We know that 2 is the prime number and cannot be factored further.
So, the prime factorization of 8 is \[2\times 2\times 2\].

(b) We need to find the prime factorization of the 20.
We know that $20=2\times 10$.
\[\Rightarrow 20=2\times 2\times 5\]. We know that 2, 5 are the prime numbers and cannot be factored further.
So, the prime factorization of 20 is \[2\times 2\times 5\].

(c) We need to find the prime factorization of the 34.
We know that $34=2\times 17$. We know that 2 and 17 are the prime numbers and cannot be factored further.
So, the prime factorization of 34 is \[2\times 17\].

(d) We need to find the prime factorization of the 44.
We know that $44=2\times 22$.
\[\Rightarrow 44=2\times 2\times 11\]. We know that 2 and 11 are the prime numbers and cannot be factored further.
So, the prime factorization of 44 is \[2\times 2\times 11\].

(e) We need to find the prime factorization of the 21.
We know that $21=3\times 7$. We know that 3 and 7 are the prime numbers and cannot be factored further.
So, the prime factorization of 21 is \[3\times 7\].

(f) We need to find the prime factorization of the 42.
We know that $42=2\times 21$.
\[\Rightarrow 42=2\times 3\times 7\]. We know that 2, 3 and 7 are the prime numbers and cannot be factored further.
So, the prime factorization of 42 is \[2\times 3\times 7\].

(g) We need to find the prime factorization of the 38.
We know that $38=2\times 19$. We know that 2 and 19 are the prime numbers and cannot be factored further.
So, the prime factorization of 38 is \[2\times 19\].

(h) We need to find the prime factorization of the 45.
We know that $45=3\times 15$.
\[\Rightarrow 45=3\times 3\times 5\]. We know that 3 and 5 are the prime numbers and cannot be factored further.
So, the prime factorization of 45 is \[3\times 3\times 5\].

Note: Whenever we get this type of problem involving factorization of even numbers, we should use the fact that every even number is divisible 2 and also 2 is only an even prime number. We can also prime factorize the numbers by using the upside-down division process. We should know that the prime number can only be divisible by 1 and itself before solving this problem. Similarly, we can expect problems to find the LCM of the numbers 45, 42 and 60.