N Is A Positive Integer. Is N The Square Of An Integer? GMAT Data Sufficiency

Question: N is a positive integer. Is n the square of an integer?

1) 4n is the square of an integer
2) n^3 is the square of an integer

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

“N is a positive integer. Is n the square of an integer?” - is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Approach Solution : 1

Statement 1: An integer's square is 4n

Given that n is an integer, and that 4 is already a perfect square of an integer, n must also be an integer.

For example, if 4n = 4, n = 1, then n is also a perfect square of an integer

For instance, if 4n = 16, n = 4 is the square of an integer, and n is also an integer

Therefore this statement is sufficient.

Statement 2: The square of an integer is n^3

If n^3 = 1, n = 1 which is square of an Integer
If n^3 = 4, n = Not an Integer
If n^3 = 9, n = Not an Integer
If n^3 = 16, n = Not an Integer
If n^3 = 25, n = Not an Integer
If n^3 = 36, n = Not an Integer
If n^3 = 49, n = Not an Integer
If n^3 = 64, n = 4 which is a perfect square

Keep in mind that the number must have the form for n to be an integer and for n^3 to be a square.

That is, n must be a perfect square

Therefore this statement is sufficient.

Correct Answer: (D)

Approach Solution : 2

If one is familiar with the properties of perfect squares the question is pretty easy

The properties are,

• They have odd numbers of distinct factors
• Even powers of prime factors, respectively.
• The factors added together are odd.
• They have an odd number of factors that are odd and an even number of factors that are even.

Statement 1: An integer's square is 4n

The square of an integer is 4n, and the perfect square (2^2) is 4.

This results in another perfect square when we multiply it by n, indicating that n is also a perfect square because the prime factor of 4 has even power, indicating that prime factors of n must also have even powers for 4n to be a perfect square

Therefore this statement is sufficient.

Statement 2: The square of an integer is n^3

It's fascinating. Revisit the roots and powers idea and borrow some formulas from it.
(A^m)^n = a^(m*n)

All of the prime factors of n^3 have even power if it is a perfect square. When an odd number is multiplied by a power of three, the result will be even if the second number is also odd.

Therefore, n is also a perfect square because n must have even powers for its prime factors to make n^3 a perfect square.

Therefore this statement is sufficient.

Correct Answer: (D)

Approach Solution : 3

There is only one variable (n) in the initial condition, so the number of equations should match. So, you only need one equation.

Statement 1: An integer's square is 4n

Since 4n is an even number, m^2 should be even in the expression 4n=m2 (where m is some integer).

Then, m^2 = (2k)^2, where k is an integer,

and since 4n = (2k)^2 = 4k^2

As a result, n=k^2

Therefore the answer is both yes and sufficient.

Statement 2: The square of an integer is n^3

The expression n=3√(t^2) is derived from n^3=t^2, where t is an integer.

The cube root should be eliminated because n is a positive integer.

This means that n=3√(t^2) = 3√({s^3}^2) = 3√(s^6) = s^2

This is also true and sufficient.

Correct Answer: (D)

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