Consider the positive integer N = 2^4 \times 3^3 \times 5^2. Which of the following statem… — State PSC Exam

Consider the positive integer $N = 2^4 \times 3^3 \times 5^2$ . Which of the following statements regarding the properties of the factors and multiples of $N$ is/are correct?

The number of positive factors of $N$ that are perfect squares is equal to 12.Answer
The sum of the reciprocals of all positive factors of $N$ is equal to $\frac{\sigma(N)}{N}$ , where $\sigma(N)$ is the sum of all positive factors of $N$ .Answer
The number of ordered pairs of positive integers $(x, y)$ such that $\text{lcm}(x, y) = N$ is equal to 315.Answer
D
The product of all positive factors of $N$ is equal to $N^{60}$ .

Answer

The statements asserting that the number of perfect square factors is 12, the sum of factor reciprocals equals

\frac{\sigma(N)}{N}

, and the number of ordered pairs

(x, y)

such that

\text{lcm}(x, y) = N

is 315 are all correct.

The statements asserting that

N

has 12 perfect square factors, that the sum of reciprocals of all factors is

\frac{\sigma(N)}{N}

, and that there are 315 ordered pairs

(x, y)

with

\text{lcm}(x, y) = N

are all mathematically correct applications of prime factorization principles.

Step-by-Step Solution

Determine the number of perfect square factors of

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

Any factor

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

is a perfect square if

a, b, c

are even. The allowed values are

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

(3 options),

b \in \{0, 2\}

(2 options), and

c \in \{0, 2\}

(2 options). Total perfect square factors =

3 \times 2 \times 2 = 12

Perfect square factors require non-negative even exponents for each prime factor.

Evaluate the sum of the reciprocals of all factors of

N

d

ranges over all divisors of

N

\frac{N}{d}

also ranges over all divisors of

N

. Therefore,

\sum_{d|N} \frac{1}{d} = \sum_{d|N} \frac{N/d}{N} = \frac{1}{N} \sum_{d'|N} d' = \frac{\sigma(N)}{N}

Pairing each factor

d

with its complementary factor

N/d

converts the reciprocal sum into the total factor sum divided by

N

Calculate the number of ordered pairs

(x, y)

such that

\text{lcm}(x, y) = N

For each prime

p_i^{e_i}

, the maximum exponent in

x

and

y

must be

e_i

. The number of choices for

(a_i, b_i)

such that

\max(a_i, b_i) = e_i

2e_i + 1

. For

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

, total ordered pairs =

(2(4)+1)(2(3)+1)(2(2)+1) = 9 \times 7 \times 5 = 315

The exponent of each prime factor in

\text{lcm}(x, y)

is determined independently by

\max(a_i, b_i) = e_i

Compute the product of all positive factors of

N

Total number of factors

D(N) = (4+1)(3+1)(2+1) = 60

. The product of all factors is

N^{D(N)/2} = N^{60/2} = N^{30}

Factors can be grouped into

D(N)/2

pairs, each multiplying to

N

Key Concept

Advanced properties of prime factorization including perfect square factor enumeration, reciprocal factor sum identity, LCM pair counting formula, and factor product relations.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic