If k is a product of the integers 6, 9, 10, and 25, what can be said about its factors?

If k is a product of the integers 6, 9, 10, and 25, what can be said about its factors?

• A. k must be a prime number
• B. k must be divisible by multiples of 450
• C. k cannot have factors greater than 25
• D. k is limited to single-digit factors

Answer: (B)

To understand why the correct answer is that k must be divisible by multiples of 450, we first need to analyze the product of the integers 6, 9, 10, and 25. Calculating the product, we find: - The prime factorization of 6 is 2 × 3. - The prime factorization of 9 is 3². - The prime factorization of 10 is 2 × 5. - The prime factorization of 25 is 5². Now, let's combine these factorizations to determine the overall prime factorization of k: k = 6 × 9 × 10 × 25 = (2 × 3) × (3²) × (2 × 5) × (5²) = 2² × 3³ × 5³. Next, we note the factorization of 450, which is: 450 = 2 × 3² × 5². To see if k is divisible by multiples of 450, we can analyze their prime factorizations. Since k includes 2², 3³, and 5³ in its factorization, it indeed contains enough of each prime

To understand why the correct answer is that k must be divisible by multiples of 450, we first need to analyze the product of the integers 6, 9, 10, and 25.

Calculating the product, we find:

• The prime factorization of 6 is 2 × 3.

• The prime factorization of 9 is 3².

• The prime factorization of 10 is 2 × 5.

• The prime factorization of 25 is 5².

Now, let's combine these factorizations to determine the overall prime factorization of k:

k = 6 × 9 × 10 × 25 = (2 × 3) × (3²) × (2 × 5) × (5²)

= 2² × 3³ × 5³.

Next, we note the factorization of 450, which is:

450 = 2 × 3² × 5².

To see if k is divisible by multiples of 450, we can analyze their prime factorizations. Since k includes 2², 3³, and 5³ in its factorization, it indeed contains enough of each prime