A number is written with the following factorization: 23 × 4 × 5. Is this factorization a prime factorization? Explain why or why not. If it is not correct, give the correct prime factorization of the number.
no.
4 is not prime
The given factorization of the number 23 × 4 × 5 is not a prime factorization. In a prime factorization, all the factors should be prime numbers, meaning they can only be divided by 1 and themselves without any remainder.
In the given factorization, the number 4 is not a prime number because it can be divided evenly by 2. To find the correct prime factorization, we need to factorize 4 into its prime factors.
The prime factorization of 4 is 2 × 2. Therefore, the correct prime factorization of the number is 23 × 2 × 2 × 5.
To determine whether the given factorization is a prime factorization, we need to check if every factor listed is a prime number.
Let's examine each factor: 23, 4, and 5.
First, we establish that 23 is indeed a prime number because it has no divisors other than 1 and itself.
Next, we check the factor 4. While 4 is not a prime number because it can be divided evenly by 2, it is still a factor of the number. However, we need to express it using only prime factors.
To do this, we can decompose 4 into its prime factors by factoring out the power of 2. 4 can be written as 2^2, where the exponent represents the number of times the prime factor is multiplied. Therefore, the prime factorization of 4 is simply 2^2.
Finally, we consider the factor 5. Since 5 is already a prime number, there is no further decomposition needed.
Thus, the given factorization of 23 × 4 × 5 is not a prime factorization because the factor 4 needs to be expressed as 2^2. The correct prime factorization of the number is 2^2 × 23 × 5.