Question

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

(a) is \\(2^1 \cdot 5^1 = 10\\) a factor of \\(a\\)? why or why not?
a. no. the prime number 3 is not a factor of 10, but it is a factor of \\(a\\).
b. yes. all the prime factors of \\(a\\) appear in 10.
c. yes. all the prime factors of 10 appear in \\(a\\) and to at least as high a power.
d. no. since \\(2^1\\) is a factor of 10, the prime number 2 appears to a higher power in 10 than it does in \\(a\\).

(b) is \\(2^4 \cdot 3^1 \cdot 5^1 = 240\\) a factor of \\(a\\)? why or why not?
a. no. since \\(2^4\\) is a factor of 240, the prime number 2 appears to a higher power in 240 than it does in \\(a\\).
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\\).

Explanation:

Analyze the prime factorization of \(a\)

We are given the number \(a\) in its prime factorized form:

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

For any integer \(d\) to be a factor of \(a\), every prime factor of \(d\) must also be a prime factor of \(a\), and its exponent in the prime factorization of \(d\) must be less than or equal to its exponent in the prime factorization of \(a\).

Evaluate part (a)

We want to determine if \(10 = 2^1 \cdot 5^1\) is a factor of \(a\).
Using the Prime Factorization Divisibility concept:

• The prime factors of \(10\) are \(2\) and \(5\).
• Both \(2\) and \(5\) are prime factors of \(a\).
• The exponent of \(2\) in \(10\) is \(1\), which is less than or equal to its exponent in \(a\) (\(1 \le 2\)).
• The exponent of \(5\) in \(10\) is \(1\), which is less than or equal to its exponent in \(a\) (\(1 \le 1\)).

Therefore, \(10\) is a factor of \(a\) because all the prime factors of \(10\) appear in \(a\) and to at least as high a power. This matches option C.

Evaluate part (b)

We want to determine if \(240 = 2^4 \cdot 3^1 \cdot 5^1\) is a factor of \(a\).
Using the Prime Factorization Divisibility concept:

• The exponent of \(2\) in \(240\) is \(4\).
• The exponent of \(2\) in \(a\) is \(2\).
• Since \(4 > 2\), the prime number \(2\) appears to a higher power in \(240\) than it does in \(a\).

Therefore, \(240\) is not a factor of \(a\). This matches option A.

Answer:

Question 1

• (A) No. The prime number 3 is not a factor of 10, but it is a factor of a.
• (B) Yes. All the prime factors of a appear in 10.
• (C) Yes. All the prime factors of 10 appear in a and to at least as high a power. (Correct answer)
• (D) No. Since \(2^1\) is a factor of 10, the prime number 2 appears to a higher power in 10 than it does in a.

Question 2

• (A) No. Since \(2^4\) is a factor of 240, the prime number 2 appears to a higher power in 240 than it does in a. (Correct answer)
• (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.