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 18.
(simplify your answer. type a whole number.)

(e) make an orderly list of all of the factors of a. choose the correct answer below.
a. the factors, in order, are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 60, 90, and 180.
b. the factors, in order, are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.
c. the factors, in order, are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 16, 18, 20, 30, 36, 45, 60, 80, 90, and 180.
d. the factors, in order, are 1, 2, 3, 4, 5, 6, 9, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.

Explanation:

Calculate the value of a

Using the Number of Factors Formula knowledge point

$$ a = 2^2 \cdot 3^2 \cdot 5^1 = 4 \cdot 9 \cdot 5 = 180 $$

Determine the total number of factors

Using the Number of Factors Formula knowledge point

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

List all factors of 180 systematically

Using the Number of Factors Formula knowledge point

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Compare with the given options

Using the Number of Factors Formula knowledge point

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Answer:

• A. The factors, in order, are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 60, 90, and 180.
• B. The factors, in order, are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180. (Correct answer)
• C. The factors, in order, are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 16, 18, 20, 30, 36, 45, 60, 80, 90, and 180.
• D. The factors, in order, are 1, 2, 3, 4, 5, 6, 9, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.