Let N be a positive integer with the prime factorization N = 2^4 \times 3^2 \times 5^3. Wh… — State PSC Exam

Let $N$ be a positive integer with the prime factorization $N = 2^4 \times 3^2 \times 5^3$ . Which of the following statements regarding the positive factors of $N$ are correct?

The total number of positive factors of $N$ that are perfect squares is $12$ .Answer
The number of positive factors of $N$ that are divisible by $15$ is $30$ .Answer
C
The total number of positive even factors of $N$ is $50$ .
D
The number of positive odd factors of $N$ is $15$ .

Answer

The correct statements are that the total number of positive factors of

N

that are perfect squares is

12

, and the number of positive factors of

N

that are divisible by

15

30

The statement asserting that

N

has

12

factors that are perfect squares is correct because choosing even powers (

0, 2, 4

for

2

;

0, 2

for

3

; and

0, 2

for

5

) yields

3 \times 2 \times 2 = 12

factors. The statement asserting that

30

factors are divisible by

15

is also correct because forcing exponents of

3

and

5

to be at least

1

yields

5 \times 2 \times 3 = 30

factors.

Step-by-Step Solution

Analyze the prime factorization

N = 2^4 \times 3^2 \times 5^3

Any positive factor of

N

is of the form

2^a \times 3^b \times 5^c

, where

0 \le a \le 4

0 \le b \le 2

, and

0 \le c \le 3

By the Fundamental Theorem of Arithmetic, factors inherit prime bases with constrained exponents.

Count the number of factors that are perfect squares

For a perfect square,

a \in \{0, 2, 4\}

(3 choices),

b \in \{0, 2\}

(2 choices), and

c \in \{0, 2\}

(2 choices). Total square factors

= 3 \times 2 \times 2 = 12

All prime exponents in a perfect square must be even non-negative integers.

Count the number of factors divisible by

15

Since

15 = 3^1 \times 5^1

a \in \{0, 1, 2, 3, 4\}

(5 choices),

b \in \{1, 2\}

(2 choices), and

c \in \{1, 2, 3\}

(3 choices). Total factors divisible by

15 = 5 \times 2 \times 3 = 30

Divisibility by

15

mandates at least one factor of

3

and at least one factor of

5

Verify even and odd factor counts to evaluate the remaining claims

Even factors require

a \ge 1

, giving

4 \times 3 \times 4 = 48

even factors. Odd factors require

a = 0

, giving

1 \times 3 \times 4 = 12

odd factors.

Evaluating

a \ge 1

a = 0

confirms that claims specifying

50

even factors and

15

odd factors are incorrect.

Key Concept

Counting factors with constrained exponents in prime factorizations

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