Answered

Get detailed and reliable answers to your questions on IDNLearn.com. Ask your questions and receive detailed and reliable answers from our experienced and knowledgeable community members.

The prime factorization of a number is [tex]3^2 \times 5^3 \times 7[/tex]. Which statement is true about the factors of the number?

A. Twenty-one is a factor of the number because both 3 and 7 are prime factors.
B. Twenty-one is not a factor of the number because 21 is not prime.
C. Ninety is a factor of the number because [tex]3^2=9[/tex] and 90 is divisible by 9.
D. Ninety is not a factor of the number because 90 is not divisible by 7.

Sagot :

GinnyAnsweridn
To determine which of the given statements about the factors of the number \( 3^2 \times 5^3 \times 7 \) is true, we can proceed step-by-step to understand the factors and the logical basis for each of the statements.

First, let’s recognize the prime factorization and compute the number itself:
[tex]\[ 3^2 \times 5^3 \times 7 \][/tex]

Now, we'll evaluate each statement individually based on the factors of the number:

### 1. Twenty-one is a factor of the number because both 3 and 7 are prime factors.
We need to check if 21 is a factor of the number. The number 21 can be factored into prime numbers as:
[tex]\[ 21 = 3 \times 7 \][/tex]

Both 3 and 7 are prime factors of the number \( 3^2 \times 5^3 \times 7 \). Since both 3 and 7 exist in the prime factorization, 21 is indeed a factor of the number. Therefore, this statement is true.

### 2. Twenty-one is not a factor of the number because 21 is not prime.
This statement is incorrect because whether 21 is prime or not is irrelevant to whether it is a factor of the number \( 3^2 \times 5^3 \times 7 \). The criterion is whether its prime factors (3 and 7) are included in the factorization of the number, which they are.

### 3. Ninety is a factor of the number because \( 3^2 = 9 \) and 90 is divisible by 9.
Let's see if 90 is a factor of the number. First, factor 90 into its prime components:
[tex]\[ 90 = 2 \times 3^2 \times 5 \][/tex]

Although \( 3^2 \) and 5 are both present in our number's factorization, the factor of 2 is missing in \( 3^2 \times 5^3 \times 7 \), making this statement incorrect.

### 4. Ninety is not a factor of the number because 90 is not divisible by 7.
Since we noted above that 90 can be expressed as \( 2 \times 3^2 \times 5 \) and our number includes a factor of 7, but 90 does not, 90 is indeed not a factor of the number. Therefore, this statement is true due to the missing factor of 7 in 90.

In conclusion:
- The statement "Twenty-one is a factor of the number because both 3 and 7 are prime factores." is true.
- The statement "Ninety is not a factor of the number because 90 is not divisible by 7." is true.

Combining these results with the numbers given:
The number computed is [tex]\( 7875 \)[/tex], [tex]\( 21 \)[/tex] is a factor, and [tex]\( 90 \)[/tex] is not a factor.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.