To determine which statement about the factors of the number is true, let's first break down the prime factorization provided:
The number can be expressed as: \[ 2^3 \times 3^2 \times 5^1 \]
Now, let's analyze each statement:
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"Fifteen is a factor of the number because both 3 and 5 are prime factors."
- The prime factorization of 15 is \( 3^1 \times 5^1 \).
- Since the number \( 2^3 \times 3^2 \times 5^1 \) contains at least \( 3^1 \) and \( 5^1 \), 15 is indeed a factor of the number.
- This statement is true.
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"Fifteen is not a factor of the number because 15 is odd and the number is even."
- The fact that 15 is odd does not determine whether it is a factor or not. Since we concluded in the first statement that 15 is indeed a factor, this statement is false.
- This statement is false.
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"Sixteen is a factor of the number because 2 cubed = 8 and 16 is divisible by 8."
- The prime factorization of 16 is \( 2^4 \). Since the number has \( 2^3 \), it does not have enough factors of 2 to create \( 2^4 \), meaning 16 cannot be a factor of the number.
- This statement is false.
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"Sixteen is not a factor of the number because the exponent of 2 is not even."
- The exponent of 2 in the prime factorization is 3 (which is odd), but the reason this statement claims 16 is not a factor is incorrectly stated. The true reason is that the exponent of 2 in the number is not high enough to factor into 16.
- This statement is misleading and ultimately false.
Given the analysis, the first statement is the only true statement:
Fifteen is a factor of the number because both 3 and 5 are prime factors.