If k is a positive integer such that k is divisible by 15 and k^2 is divisible by 360, wha… — GRE General Test

If $k$ is a positive integer such that $k$ is divisible by $15$ and $k^2$ is divisible by $360$ , what is the minimum possible number of positive divisors of $k$ ?

A
6
B
8
12Answer
D
18
E
24

Answer

The minimum possible number of positive divisors of

k

is 12.

The minimum possible number of positive divisors occurs when

k

has the smallest possible prime exponents satisfying all divisibility conditions. Since

15 = 3 \times 5

divides

k

k

must have prime factors

3^1

and

5^1

. For

k^2

to be divisible by

360 = 2^3 \times 3^2 \times 5^1

k^2

must have at least

2^3

, which requires

k

to have at least

2^2 = 4

. Thus, the minimal prime factorization of

k

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

. Using the divisor count formula

(e_1 + 1)(e_2 + 1)(e_3 + 1)

, we get

(2+1)(1+1)(1+1) = 12

positive divisors.

Step-by-Step Solution

Find the prime factorizations of 15 and 360.

15 = 3^1 \times 5^1

and

360 = 2^3 \times 3^2 \times 5^1

Expressing numbers in terms of prime factors reveals the required prime exponents for divisibility.

Determine the minimum exponents of the prime factors required for

k

Since

15

divides

k

k

must contain at least

3^1

and

5^1

. For

k^2

to be divisible by

360 = 2^3 \times 3^2 \times 5^1

k^2

must contain at least

2^3

, which means

k

must contain at least

2^2

. The required factors

3^2

and

5^1

k^2

are automatically provided since

(3^1)^2 = 3^2

and

(5^1)^2 = 5^2

. Thus, the minimal

k

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

Taking the minimum required exponent for each prime factor minimizes the total number of divisors of

k

Calculate the number of positive divisors of the minimal

k

The number of divisors for

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

(2 + 1)(1 + 1)(1 + 1) = 3 \times 2 \times 2 = 12

The divisor count formula adds 1 to each prime exponent and multiplies the results.

Key Concept

Properties of Integer Divisibility and Divisor Counting

Estimated Time:1m 30s

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