A number is written with the following factorization: 22 × 3 × 54 × 8 × 112. Is this factorization a prime factorization? Explain why or why not. If it is not correct, give the correct prime factorization of the number.
If you mean 2^2 * 3 * 5^4 * 11^2
then yes, since each factor is a power of as prime
As you wrote it, no, since 22, 54, 112 are not primes.
but the 8 ?
8 = 2*2*2 = 2^3
Thus 8 is itself not prime
To determine if the factorization is a prime factorization, we need to check if all the factors are prime numbers.
We can do this by checking each factor in the factorization individually:
22 is not a prime number because it can be factored into 2 × 11.
3 is a prime number.
54 is not a prime number because it can be factored into 2 × 3 × 3 × 3.
8 is not a prime number because it can be factored into 2 × 2 × 2.
112 is not a prime number because it can be factored into 2 × 2 × 2 × 2 × 7.
Since each factor in the given factorization can be further factored, the factorization is not a prime factorization.
To find the correct prime factorization of the number, we need to factorize each factor into prime numbers:
22 = 2 × 11
3 = 3
54 = 2 × 3 × 3 × 3 = 2 × (3^3)
8 = 2 × 2 × 2 = 2^3
112 = 2 × 2 × 2 × 2 × 7 = 2^4 × 7
Now, we can combine all the prime factors together to get the correct prime factorization:
22 × 3 × 54 × 8 × 112 = 2 × 11 × 3 × (2 × 3^3) × (2^3) × (2^4 × 7)
Simplifying the expression:
= 2 × 11 × 3 × 2 × 3 × 3 × 2 × 2 × 2 × 2 × 7
= 2^8 × 3^4 × 7 × 11
So, the correct prime factorization of the given number is 2^8 × 3^4 × 7 × 11.