A positive integer n has the prime factorization n = 2^a \cdot 3^b \cdot 5^c, where a, b, … — GRE General Test

A positive integer $n$ has the prime factorization $n = 2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are positive integers. If $n$ is divisible by $36$ and is a divisor of $5{}400$ , which of the following values could be the total number of positive divisors of $n$ ? Select all such values.

18Answer
B
20
27Answer
32Answer
E
40

Answer

18, 27, and 32 are all possible total numbers of positive divisors for

n

Prime factorization gives

36 = 2^2 \cdot 3^2

and

5{}400 = 2^3 \cdot 3^3 \cdot 5^2

. For

n = 2^a \cdot 3^b \cdot 5^c

to be a multiple of 36 and a divisor of 5,400 with positive integer exponents,

a \in \{2,3\}

b \in \{2,3\}

, and

c \in \{1,2\}

. The total number of divisors is

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

. The possible values for this product are 18, 24, 27, 32, 36, and 48. Among the choices, 18, 27, and 32 are valid values.

Step-by-Step Solution

Find the prime factorizations of the boundary numbers 36 and 5,400.

36 = 2^2 \cdot 3^2

and

5{}400 = 2^3 \cdot 3^3 \cdot 5^2

Establishing the prime factor bounds determines the range of possible values for exponents

a

b

, and

c

Determine the constraints on exponents

a

b

, and

c

Since

36 \mid n

a \ge 2

b \ge 2

, and

c \ge 1

(given

c

is a positive integer). Since

n \mid 5{}400

a \le 3

b \le 3

, and

c \le 2

. Thus,

a \in \{2, 3\}

b \in \{2, 3\}

, and

c \in \{1, 2\}

Divisibility rules require prime factor exponents of a multiple to be greater than or equal to those of the divisor, and exponents of a divisor to be less than or equal to those of the multiple.

Calculate all possible total divisor counts using the formula

d(n) = (a+1)(b+1)(c+1)

Possible factor values are

(a+1) \in \{3, 4\}

(b+1) \in \{3, 4\}

, and

(c+1) \in \{2, 3\}

. Evaluating all combinations yields:

3 \cdot 3 \cdot 2 = 18

3 \cdot 3 \cdot 3 = 27

3 \cdot 4 \cdot 2 = 24

3 \cdot 4 \cdot 3 = 36

4 \cdot 4 \cdot 2 = 32

, and

4 \cdot 4 \cdot 3 = 48

The total number of positive integer divisors is found by adding 1 to each exponent in the prime factorization and multiplying the results.

Compare the calculated divisor counts with the options provided.

The values 18, 27, and 32 appear in the calculated set of possible total divisors.

Direct matching identifies all valid options.

Key Concept

Prime factor exponent bounds and total number of positive divisors formula

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