The prime factorization of a positive integer N is of the form 2^a \times 3^b \times 5^c, … — ACT

The prime factorization of a positive integer $N$ is of the form $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are non-negative integers. The number $N$ has exactly $12$ positive factors. If $N$ is a multiple of $4$ but is not divisible by $3$ , what is the smallest possible value of $N$ ?

Answer: 160

Answer

The smallest possible value of

N

is 160.

The correct value is 160 because it satisfies all the conditions: its prime factorization has only 2 and 5 (no 3, so not divisible by 3), the exponent of 2 is 5 (which is greater than or equal to 2, so it is a multiple of 4), the number of factors is

(5+1)(1+1) = 12

, and it is the smallest such integer.

Step-by-Step Solution

Set up the factor counting formula using the prime factorization.

The number of positive factors of

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

(a+1)(b+1)(c+1) = 12

The number of positive factors of any integer is found by adding one to each exponent in its prime factorization and multiplying the results.

Apply the given divisibility conditions to simplify the exponents.

Since

N

is not divisible by

3

b = 0

. Since

N

is a multiple of

4

a \ge 2

. The equation simplifies to

(a+1)(c+1) = 12

with

a+1 \ge 3

A number is not divisible by a prime if its exponent in the prime factorization is 0. A number is a multiple of 4 if the exponent of 2 in its prime factorization is at least 2.

List all possible pairs of

(a+1, c+1)

that multiply to 12 where

a+1 \ge 3

, and compute the corresponding values of

N

Case 1:

a+1 = 3 \implies a = 2, c = 3 \implies N = 2^2 \times 5^3 = 500

. Case 2:

a+1 = 4 \implies a = 3, c = 2 \implies N = 2^3 \times 5^2 = 200

. Case 3:

a+1 = 6 \implies a = 5, c = 1 \implies N = 2^5 \times 5^1 = 160

. Case 4:

a+1 = 12 \implies a = 11, c = 0 \implies N = 2^{11} \times 5^0 = 2048

Testing all possible divisor pairs of 12 that satisfy the constraint on

a

allows us to find all possible candidate values for

N

Determine the smallest value of

N

from the candidate values.

The smallest candidate is 160.

Comparing the values 500, 200, 160, and 2048 shows that 160 is the minimum.

Key Concept

Factors, Multiples, and Prime Factorization

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