The prime factorization of a number is 2 cubed times 3 squared times 5. Which is a true statement about the factors of the number?

Question

Fifteen is a factor of the number because both 3 and 5 are prime factors.
Fifteen is not a factor of the number because 15 is odd and the number is even.
Sixteen is a factor for the number because 2 cubed = 8 and 16 is divisible by 8.
Sixteen is not a factor of the number because the exponent of 2 is not even.

Answer 1

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:

1. "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.

2. "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.

3. "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.

4. "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.