If 0 < x < 53, What is the Value of Integer x?

Question: If 0 < x < 53, what is the value of integer x?

1. x is divisible by atleast 2 prime numbers greater than 2
2. \(\sqrt{x+1}-1\)is prime

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.

“If 0 < x < 53, what is the value of integer x?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". 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.

Answer

Approach 1

Firstly, consider statement (1), we have: x is divisible by atleast 2 prime numbers greater than 2. This statement is insufficient in itself.

Now considering statement (2), we have \(\sqrt{x+1}-1\) is prime. This statement is also insufficient in itself. Run squares of 2, 3, 4, 5, 6, and 7.

In actual, we need only squares of 2, 4, 6 as x – 1 is prime and thus, the square root result has to be even to make x – 1 odd.

If x = 15 \(\sqrt{16}=4\), and 4 – 1 = 3

Also, if x = 35, \(\sqrt{36}=6\) , and 6 – 1 = 5. There are several values that work for x.

S (1) and S (2) combined are also insufficient. If we take both the statements together, x can be 35 or 15. Both are divisible by several primes greater than 2.

Correct option: E

Approach 2

Let’s simplify the statement (2), we have:

\(\sqrt{x+1}-1\)= prime

\(\sqrt{x+1}\)= prime + 1

Squaring on both sides, we get:
x + 1 = \((prime + 1)^2\)

x +1 = \(prime^2\)+ 2 prime + 1
x = \(prime^2\)+ 2 prime

Choose prime as follows
Let, prime = 2, then x = 8
Let prime = 3, then x = 15
Let prime = 5, then x = 35

We can put prime = 7, because this will violate the question that 0 < x < 53
Now taking statement (1), we have x is divisible by atleast 2 prime numbers greater than 2. This statement is insufficient in itself.
Combining statement (1) and (2), we have:

x = 15 that has two prime numbers 3, 5
or
x = 35 that has two prime numbers 5, 7

Correct option: E

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