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

The prime factorization of a positive integer $N$ is $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are positive integers. If $N$ has exactly $12$ distinct positive factors, what is the least possible value of $N$ ?

Answer: 60

Answer

The number of distinct positive factors of

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

is given by the formula

(a+1)(b+1)(c+1)

. Given that

N

has exactly

12

positive factors and

a, b, c

are positive integers (each at least

1

), we must factor

12

into three integers that are each at least

2

. The only way to do this is

2 \times 2 \times 3

. This means the set of exponents

\{a, b, c\}

must be

\{1, 1, 2\}

. To minimize the value of

N

, we pair the largest exponent

2

with the smallest prime base

2

, and the exponents of

1

with the prime bases

3

and

5

. This yields the minimum value

2^2 \times 3^1 \times 5^1 = 60

Step-by-Step Solution

Determine the formula for the number of positive factors of

N

The number of positive factors is

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

For any integer expressed as a product of prime powers, the number of positive factors is the product of each exponent increased by 1.

Find the possible values for the exponents

a

b

, and

c

Since

a, b, c \ge 1

, we must have

a+1 \ge 2

b+1 \ge 2

, and

c+1 \ge 2

. The only factorization of

12

into three integers each at least

2

2 \times 2 \times 3

. Thus, the exponents

\{a, b, c\}

must be

\{1, 1, 2\}

in some order.

We must find a set of integer values for the exponents that satisfy the factor count constraint.

Minimize the value of

N

by assigning exponents to the prime bases.

Assign the largest exponent (

2

) to the smallest base (

2

), and the smaller exponents (

1

) to the larger bases (

3

and

5

). This gives

a=2, b=1, c=1

, so

N = 2^2 \times 3^1 \times 5^1 = 60

To minimize the total product, the largest power must be applied to the smallest base.

Key Concept

Finding the number of factors of a positive integer using its prime factorization.

Estimated Time:1m 15s

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