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 $72$ ?

Answer: 21

Answer

The correct answer is 21. By analyzing the exponent bounds of

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

, there are 48 factors of

N

divisible by

12

(

a \ge 2, b \ge 1

). Among these, 27 factors are also divisible by

72

(

a \ge 3, b \ge 2

). Subtracting the two counts gives

48 - 27 = 21

factors divisible by 12 but not by 72.

Step-by-Step Solution

Represent the general form of a factor of

N

Every positive factor of

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

can be uniquely written as

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

, where

a \in \{0, 1, 2, 3, 4, 5\}

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

, and

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

By the Fundamental Theorem of Arithmetic, any factor of a number in prime-factored form takes exponents bounded by the original number's exponents.

Identify exponent constraints for divisibility by 12

Since

12 = 2^2 \times 3^1

, for

f

to be divisible by

12

, we must have

a \ge 2

and

b \ge 1

Divisibility requires prime factor exponents in the factor to be at least as large as those in the divisor.

Calculate the total number of factors divisible by 12

Number of choices for

a

5 - 2 + 1 = 4

(values

2, 3, 4, 5

). Number of choices for

b

4 - 1 + 1 = 4

(values

1, 2, 3, 4

). Number of choices for

c

2 - 0 + 1 = 3

(values

0, 1, 2

). Total count =

4 \times 4 \times 3 = 48

By the multiplication principle of counting, independent choices for each exponent are multiplied.

Identify exponent constraints and count factors divisible by 72

Since

72 = 2^3 \times 3^2

, a factor is divisible by

72

a \ge 3

and

b \ge 2

. Number of choices:

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

(3 choices),

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

(3 choices),

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

(3 choices). Total count =

3 \times 3 \times 3 = 27

Every factor of

N

that is divisible by 72 is inherently also a factor divisible by 12, forming a strict subset.

Apply the principle of subtraction to find the required count

Required count = (Factors divisible by 12) - (Factors divisible by 72) =

48 - 27 = 21

The set of factors divisible by 12 but not by 72 is the set difference between factors divisible by 12 and factors divisible by 72.

Key Concept

Counting Factors with Divisibility Constraints using Prime Factorization

Estimated Time:2m 0s

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