Is the positive integer n divisible by 36? Statement I: n^2 is divisible by 108. Statement… — State PSC Exam

Is the positive integer $n$ divisible by $36$ ?

Statement I: $n^2$ is divisible by $108$ .
Statement II: $n^3$ is divisible by $576$ .

Which of the following options correctly describes the sufficiency of the statements?

A
Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.
B
Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.
C
Either Statement I alone or Statement II alone is sufficient to answer the question.
D
Neither Statement I nor Statement II is sufficient to answer the question, and additional data is needed.
Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.Cevap

Cevap

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating both statements together shows that Statement I requires

n

to be a multiple of

18

(

2^1 \times 3^2

) and Statement II requires

n

to be a multiple of

12

(

2^2 \times 3^1

). The least common multiple of

18

and

12

36

, which proves that

n

is guaranteed to be divisible by

36

. Neither statement alone provides both prime factor requirements.

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Analyze Statement I individually

Statement I states that

n^2

is divisible by

108 = 2^2 \times 3^3

. In the prime factorization of a square

n^2

, all exponents must be even numbers. Thus,

n^2

must contain at least

2^2

and

3^4

. Taking square roots,

n

must be a multiple of

2^1 \times 3^2 = 18

. If

n = 18

n

is NOT divisible by

36

. If

n = 36

n

IS divisible by

36

. Because we get both 'No' and 'Yes' answers, Statement I alone is NOT sufficient.

We must test if Statement I uniquely determines whether

n

is divisible by

36

Analyze Statement II individually

Statement II states that

n^3

is divisible by

576 = 2^6 \times 3^2

. In the prime factorization of a cube

n^3

, all exponents must be multiples of

3

. Thus,

n^3

must contain at least

2^6

and

3^3

. Taking cube roots,

n

must be a multiple of

2^2 \times 3^1 = 12

. If

n = 12

n

is NOT divisible by

36

. If

n = 36

n

IS divisible by

36

. Because we get both 'No' and 'Yes' answers, Statement II alone is NOT sufficient.

We must test if Statement II uniquely determines whether

n

is divisible by

36

Combine Statement I and Statement II

From Statement I,

n

contains at least

3^2

in its prime factorization. From Statement II,

n

contains at least

2^2

in its prime factorization. Combining these requirements,

n

must contain at least

2^2 \times 3^2 = 36

. Therefore,

n

is guaranteed to be divisible by

36

. Both statements together yield a definitive 'Yes'.

Evaluating both statements together combines the minimal necessary powers of each prime factor.

Anahtar Kavram

Data Sufficiency evaluation of prime factor exponents and divisibility rules

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