Let n be a positive integer such that n is divisible by 12 and n^2 is divisible by 270. Wh… — GRE General Test

Let $n$ be a positive integer such that $n$ is divisible by $12$ and $n^2$ is divisible by $270$ . Which of the following statements MUST be true? Select all such statements.

$n$ is divisible by $180$ .Answer
$n^2$ is divisible by $10,800$ .Answer
C
$n$ is divisible by $24$ .
D
$n^3$ is divisible by $250,000$ .
$n$ has at least $18$ positive divisors.Answer

Answer

The statements asserting that

n

is divisible by

180

n^2

is divisible by

10,800

, and

n

has at least

18

positive divisors MUST be true.

Because

12 = 2^2 \cdot 3^1

divides

n

n

contains at least two factors of

2

and one factor of

3

. Because

270 = 2^1 \cdot 3^3 \cdot 5^1

divides

n^2

n^2

must contain at least three factors of

3

and one factor of

5

, meaning

n

itself must contain at least two factors of

3

and one factor of

5

. Therefore,

n

must be a multiple of

2^2 \cdot 3^2 \cdot 5^1 = 180

. Consequently: (1)

n

is divisible by

180

; (2)

n^2

is a multiple of

180^2 = 32,400

, which is divisible by

10,800

; (3)

n

has at least as many divisors as

180

, which has

(2+1)(2+1)(1+1) = 18

positive divisors.

Step-by-Step Solution

Determine the prime factorization constraints on

n

from the given divisibility conditions.

Since

12 = 2^2 \cdot 3^1

divides

n

v_2(n) \ge 2

and

v_3(n) \ge 1

. Since

270 = 2^1 \cdot 3^3 \cdot 5^1

divides

n^2

v_2(n^2) \ge 1

v_3(n^2) \ge 3

, and

v_5(n^2) \ge 1

Divisibility requires that the exponent of each prime factor in the divisor is less than or equal to the exponent in the dividend.

Deduce the minimal exponents of primes in

n

v_2(n) \ge 2

2 \cdot v_3(n) \ge 3 \implies v_3(n) \ge 2

, and

2 \cdot v_5(n) \ge 1 \implies v_5(n) \ge 1

. Thus,

n

must be a multiple of

2^2 \cdot 3^2 \cdot 5^1 = 180

Exponents of prime factors in integer factorizations must be non-negative integers.

Evaluate each statement against the minimum required value

n = 180 = 2^2 \cdot 3^2 \cdot 5^1

180 \mid n

is true.
2.

n^2

is a multiple of

180^2 = 32,400

, which is divisible by

10,800

(

32,400 = 3 \cdot 10,800

). True.
3.

180

is not divisible by

24

. False.
4.

n^3

has

5^3

, whereas

250,000

requires

5^5

. False.
5. The number of positive divisors of

180

(2+1)(2+1)(1+1) = 18

. Any multiple of

180

has at least

18

positive divisors. True.

Testing the baseline integer

n = 180

identifies necessary conditions and counterexamples.

Key Concept

Properties of Integers and Divisibility

Question

Answer

Step-by-Step Solution

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