Is the integer x a 3-digit integer? (1) x is the square of an integer

Target question: Is the integer x a 3-digit integer?

Statement 1: x is the square of an integer.
Let's TEST some values.
There are several values of x that satisfy statement 1. Here are two:
Case a: x = 10² = 100. In this case, the answer to the target question is YES, x IS a 3-digit integer
Case b: x = 9² = 81. In this case, the answer to the target question is NO, x is NOT a 3-digit integer
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 90 < x < 150
TEST some cases.
Case a: x = 100. In this case, the answer to the target question is YES, x IS a 3-digit integer
Case b: x = 91. In this case, the answer to the target question is NO, x is NOT a 3-digit integer
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are THREE values of x that satisfy BOTH statements: x = 100, x = 121 and x = 144
In all three cases, the answer to the target question is the same: YES, x IS a 3-digit integer
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

Video solution from Quant Reasoning:
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From S1:

x is the square of an integer.
Clearly Insufficient.

From S2:

90 < x < 150
X can 91 or 100
Hence, Insufficient.

Combining both:

9^2 = 81
10^2 = 100
But, 81 doesn't lie between 90 < x < 150.
Hence X will be a 3 digit integer.
Sufficient.

C is the answer.

#1
x can be single, double or triple digit insufficient
#2
90 < x < 150
double or triple digit
insufficient
from 1 &2
x=100 only possible for 10 , in range >90
IMO C

Hi All,

We're told that X is an integer. We're asked if X is a 3-digit integer. This question can be solved by TESTing VALUES.

(1) X is the SQUARE of an INTEGER.

Fact 1 tells us that X is a 'perfect square.' For example, X could be 1, 4, 9.....100, 121, 144....10,000, etc.
IF....
X = 1, then the answer to the question is NO.
X = 100, then the answer to the question is YES.
Fact 1 is INSUFFICIENT

(2) 90 < X < 150

Fact 2 gives us a range of possible values for X.
IF....
X = 91, then the answer to the question is NO.
X = 100, then the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, we know...
X is the SQUARE of an INTEGER.
90 < X < 150

There are only a few perfect squares in this given range: 100, 121 and 144. Regardless of which value X actually is, the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Hola amigos

Is the integer x a 3-digit integer?

1. \(x\) is the square of an integer.
if \(x = 1\), then \(x^2 = 1\), the answer is NOT
if \(x = 10\), then \(x^2 = 100\), the answer is YES
Insufficient

2. 90 < \(x\) < 150
if \(x = 95\), then the answer is NOT
if \(x = 145\), then the answer is YES
Insufficient

1 + 2. \(x\) can be any integer from \(91\) to \(149\). if \(x = 91\), then \(x^2 = 91*91\). \(91*91\) is greater than \(90*90 = 8100\), thus it is not a 3-digit integer. Hence at any value greater then \(91\), \(x\) will not be a 3-digit integer.
Sufficient. C

We need to determine whether integer x is a 3-digit integer.

Statement One Alone

x is the square of an integer.

Even though we know that x is a square, we cannot determine whether it’s a 3-digit integer. For example, if x = 10^2 = 100, then x is a 3-digit integer. However, if x = 5^2 = 25, then it is NOT a 3-digit integer.

Statement one alone is not sufficient to answer the question.

Statement Two Alone:

90 < x < 150

Since x could be either a 2-digit integer (e.g., 95) or a 3-digit integer (e.g., 105), statement two alone is not sufficient to answer the question.

Statements One and Two Together:

The perfect squares between 90 and 150 are 100, 121, and 144. Thus, we see that x must be a 3-digit integer.

Answer: C
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Solution

Steps 1 & 2: Understand Question and Draw Inferences

In this question, we are given

• The number x is an integer

We need to determine

• Whether x is a 3-digit integer or not.

To determine whether x is a 3-digit integer or not, we need to know the value of x, or any specific range of x from which a unique value of number of digits of x can be determined.
With this information, let’s now analyse the individual statements.

Step 3: Analyse Statement 1

As per the information given in statement 1, x is the square of an integer.
From this statement, it is not possible to determine the number of digits of x.

• If x = 22 = 4, then x is a single-digit integer.
• If x = 52 = 25, then x is a double-digit integer.
• If x = 122 = 144, then x is a three-digit integer.

Hence, statement 1 is not sufficient to answer the question.

Step 4: Analyse Statement 2

As per the information given in statement 2, 90 < x < 150.

In this given range, there can be two possibilities.

• For 90 < x < 100:

o The integer x has two digits.
• For 100 ≤ x < 150:

o The integer x has three digits.

As we cannot determine whether x is a 3-digit integer or not, statement 2 is not sufficient to answer the question.

Step 5: Combine Both Statements Together (If Needed)

From statements 1 and 2 together, we get

• The integer x is the square of an integer.
• Also, 90 < x < 150

Now, in the given range, the possible square numbers are 100, 121 and 144.

• x can be any one of these 3 values.
• Also, all these 3 numbers have 3 digits, implies x is a 3-digit integer

.

Hence, the correct answer choice is option C.


_________________

I said E originally.

I am confused about the first statement. Can someone more explicitly explain what a "square of an integer" means? In my mind initially, I thought that meant that x was any integer squared. so y^2 = x. Infinite options, so insufficient.

Then for combining the two statements is where I messed up because I'm confused about the "square of an integer". Is the "square" the result of multiplying a number by itself, or is it the value of the number being multiplied by itself?

For instance, is 9 a "square of" 81, or is 81 a "square of" 9?

The math jargon is throwing me off here.

Hi sydlyisaacs,

It might help to think of "square-ing" something in the Geometric sense... for example, a 3x3 square has an area of 9. Thus, 9 is 3^2 (re: three squared).

With Fact 1 in this prompt, we're told that X is the SQUARE of an INTEGER, so X could be 1, 4, 9, 16..... 100, 121, 144....10,000, etc.

GMAT assassins aren't born, they're made,
Rich

Answer: Option C

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