Question

let \\(a = 2^2 \cdot 3^2 \cdot 5^1\\). complete parts (a) through (e) below.

b. yes. all the prime factors of a appear in 240.
c. yes. all the prime factors of 240 appear in a and to at least as high a power.
d. no. the prime number 2 is not a factor of 10, but it is a factor of a.

(c) one factor of a is \\(b = 2^1 \cdot 3^2\\). what is the quotient when a is divided by b?
the quotient is \\(2^1 \times 3^0 \times 5^1\\).

(d) how many different factors does a possess?
the number of factors is
(simplify your answer. type a whole number.)

Explanation:

Identify the prime factorization of a

Using the Prime Factorization Divisibility knowledge point

$$ a = 2^2 \cdot 3^2 \cdot 5^1 $$

Apply the number of factors formula

To find the total number of different factors of a number from its prime factorization \(p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k}\), we use the formula:

$$ N = (e_1 + 1)(e_2 + 1) \cdots (e_k + 1) $$

For \(a = 2^2 \cdot 3^2 \cdot 5^1\), the exponents are \(e_1 = 2\), \(e_2 = 2\), and \(e_3 = 1\).

Calculate the total number of factors

Using the Quotient of Prime Factorizations knowledge point

$$ N = (2 + 1)(2 + 1)(1 + 1) = 3 \cdot 3 \cdot 2 = 18 $$

Answer:

The number of factors is <blank>18</blank>