Let N = 2^5 \times 3^4 \times 5^2. How many positive integer factors of N are divisible by… — State PSC Exam

Let $N = 2^5 \times 3^4 \times 5^2$ . How many positive integer factors of $N$ are divisible by $12$ but not divisible by $15$ ?

16Answer
B
32
C
48
D
24

Answer

16 positive factors

The number

N

is given by

2^5 \times 3^4 \times 5^2

. A factor

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

is divisible by

12 = 2^2 \times 3^1

a \in \{2,3,4,5\}

(4 choices),

b \in \{1,2,3,4\}

(4 choices), and

c \in \{0,1,2\}

(3 choices), giving

4 \times 4 \times 3 = 48

factors. Among these, those also divisible by

15 = 3^1 \times 5^1

must be divisible by

\text{LCM}(12,15) = 60 = 2^2 \times 3^1 \times 5^1

, which restricts

c \in \{1,2\}

(2 choices), yielding

4 \times 4 \times 2 = 32

factors. Subtracting these gives

48 - 32 = 16

factors.

Step-by-Step Solution

Express

N

in prime factorized form and define the general structure of its factors.

Any positive factor of

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

has the form

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

, where

0 \le a \le 5

0 \le b \le 4

, and

0 \le c \le 2

Prime factorization determines all possible divisors of a composite number.

Calculate the total number of factors divisible by

12 = 2^2 \times 3^1

For

k

to be divisible by 12, we require

a \ge 2

(

a \in \{2,3,4,5\}

, 4 choices),

b \ge 1

(

b \in \{1,2,3,4\}

, 4 choices), and

c \ge 0

(

c \in \{0,1,2\}

, 3 choices). Total factors =

4 \times 4 \times 3 = 48

A factor contains another number as a divisor if all prime exponent lower bounds of the divisor are satisfied.

Calculate the number of factors divisible by both

12

and

15

, which is equivalent to being divisible by

\text{LCM}(12, 15) = 60 = 2^2 \times 3^1 \times 5^1

For

k

to be divisible by 60, we require

a \ge 2

(4 choices),

b \ge 1

(4 choices), and

c \ge 1

(

c \in \{1,2\}

, 2 choices). Total factors =

4 \times 4 \times 2 = 32

To exclude factors divisible by 15, we must find the intersection of multiples of 12 and multiples of 15.

Subtract the factors divisible by 60 from the factors divisible by 12.

Factors divisible by 12 but not by 15 =

48 - 32 = 16

Applying the principle of set difference yields the exact count satisfying both conditions.

Key Concept

Counting Factors with Prime Factorization Constraints

Estimated Time:2m 0s

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