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.
