To analyze the given statements, let's first determine the prime factorization of the number based on the description:
The number has a prime factorization of \(3^2 \times 5^3 \times 7^1\).
From this, we can derive the factors of this number:
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Calculating 21: The number 21 can be factored into its primes: \[ 21 = 3^1 \times 7^1 \] Since both 3 and 7 are prime factors of our number, this means that 21 is indeed a factor of the number.
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Calculating 90: The number 90 can be factored as follows: \[ 90 = 2^1 \times 3^2 \times 5^1 \] To determine if 90 is a factor of our number, we need to check if all primes in 90 exist in the factorization of our number \( (3^2 \times 5^3 \times 7^1) \).
- For \(2^1\): The original number has no factor of 2, so 90 cannot be a factor of the number.
- Though it has \(3^2\) and \(5^1\) which meet the criteria, the absence of \(2^1\) in the factorization means that 90 cannot be a factor.
Now let's evaluate the statements:
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"Twenty-one is a factor of the number because both 3 and 7 are prime factors."
- This statement is true.
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"Twenty-one is not a factor of the number because 21 is not prime."
- This statement is false. Whether 21 is prime or not does not affect its status as a factor because it is made from the number's prime factors.
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"Ninety is a factor of the number because 3 squared = 9 and 90 is divisible by 9."
- This statement is false. While 90 is divisible by 9, it also requires the factorization of 2, which is missing.
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"Ninety is not a factor of the number because 90 is not divisible by 7."
- This statement is true, but it's not the complete reason, as the lack of factor \(2^1\) also confirms that 90 cannot be a factor.
The true statement to highlight is that Twenty-one is a factor of the number because both 3 and 7 are prime factors.
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