A logistics company assigns identification codes to container shipments based on a positiv… — State PSC Exam

A logistics company assigns identification codes to container shipments based on a positive integer $N$ . The number $N$ has exactly $24$ positive factors, and its prime factorization contains only the prime factors $2$ , $3$ , and $7$ . If $N$ is a multiple of $14$ and the number of odd positive factors of $N$ is $6$ , what is the smallest possible value of $N$ ?

$504$ Answer
B
$1176$
C
$252$
D
$1512$

Answer

The smallest possible value of

N

504

The number

504

has prime factorization

2^3 \times 3^2 \times 7^1

. Total positive factors =

(3+1)(2+1)(1+1) = 24

, and odd positive factors =

(2+1)(1+1) = 6

. It is divisible by

14

(

504 = 14 \times 36

) and is the smaller of the two valid numbers (

504

and

1176

Step-by-Step Solution

Express

N

in terms of its prime factors.

N = 2^a \times 3^b \times 7^c

, where

a \ge 1, b \ge 1, c \ge 1

N

contains prime factors

2, 3,

and

7

, and is a multiple of

14 = 2 \times 7

The question specifies that

N

is divisible by

14

and contains prime factors

2

3

, and

7

Formulate equations for total factors and odd factors.

Total positive factors

T(N) = (a + 1)(b + 1)(c + 1) = 24

. Odd positive factors depend only on odd prime powers (

3^b \times 7^c

), so

(b + 1)(c + 1) = 6

Odd factors correspond to choosing

2^0 = 1

for the even prime factor.

Solve for exponent

a

(a + 1) \times 6 = 24 \implies a + 1 = 4 \implies a = 3

Dividing total factors by odd factors isolates the term

(a + 1)

Find possible values for

b

and

c

Since

(b + 1)(c + 1) = 6

with

b, c \ge 1

, the possible factor pairs for

(b+1, c+1)

are

(2, 3)

(3, 2)

. Case 1:

b=1, c=2 \implies N = 2^3 \times 3^1 \times 7^2 = 1176

. Case 2:

b=2, c=1 \implies N = 2^3 \times 3^2 \times 7^1 = 504

Integer factorizations of

6

give two distinct exponent assignments.

Determine the minimum value.

Comparing

504

and

1176

, the minimum value is

504

Assigning the larger exponent to the smaller prime base (

3

instead of

7

) minimizes the product.

Key Concept

Factors and Prime Factorization Constraints

Estimated Time:2m 0s

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