If n is a positive integer and k = 5.1*10^n, what is the value of k?

Target question: What is the value of k?

Given: n is a positive integer, and k = (5.1)x(10^n)
IMPORTANT: This since n can be ANY positive integer, there are several possible values of k.
They are: 51, 510, 5100, 51000, 510000, etc

Statement 1: 6,000 < k < 500,000
If we examine the possible values of k (51, 510, 5100, 51000, 510000, etc ), we can see that only ONE value (51,000) lies within the range defined by the inequality.
So, k must equal 51,000
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: k² = 2.601 x 10^9
If k²= 2.601 x 10^9, then EITHER k = √(2.601 x 10^9) OR k = -√(2.601 x 10^9). So, it appears that we cannot answer the target question.
HOWEVER, the question also tells us that k = 5.1 x 10^n, and since 5.1 x 10^n will always have a POSITIVE value, we know that k must be POSITIVE.
If k is POSITIVE, then k ≠ -√(2.601 x 10^9)
This means that k must equal √(2.601 x 10^9)
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer =

Cheers,
Brent

Given: k = 5.1 * 10^n
n is a positive integer which means it is at least 1. Hence k's value can be anything from 51 to 5100000000...infinite zeroes

Statement 1: 6,000<k<500,000
So 5,100 is not in this range. 51,000 is in this range. 510,000 is not in this range. Therefore, k must be 51,000. Sufficient.

Statement 2: \(k^2= 2.601*10^9\)

\(k^2 = 2.601 * 10^9 = 2601 * 10^6\)

Since k is an integer, we are certain that 2601 must be the square of an integer. It is the square of 51.

If \(k^2 = 2601 * 10^6,\) \(k = 51*10^3\) \(OR -51*10^3\)
Since from the question stem, we know that k must be positive,
k should be \(51*10^3 = 51,000\). Sufficient.

Answer (D)
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If n is a positive integer and \(k=5.1*10^n\), what is the value of k?

As n is a positive integer then k could be: 51, 510, 5,100, 51,000, ... Also note that no matter the value of n, k is always a positive number.

(1) \(6,000 < k < 500,000\). Only one number from above is in this range: 51,000. Sufficient.

(2) (2) \(k^2 = 2.601 * 10^9\). We can solve quadratics to get two values of k positive and negative, but since given that k is positive then only the positive value of k will be valid. Sufficient.

Answer: D.
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Actually , i missed the given fact that k is +ve and considered the -ve value as well for the second statement.
But the Q stem itself says that k=5.1 x 10^n.

Yes, for most DS questions, before you move on to the statements, break down the question stem and analyze it properly. It should be very clear in your head what data the stem has given you and what has it asked you.
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Hi Bunuel,

If we weren't told that "n" is positive, wouldn't B still be sufficient? Isn't it true that whenever we take a square root, we always choose the positive value. Meaning, if we take a square root of 4, isn't the answer 2 and not +/- 2?

How is that any different than the k^2 value given above? Is it because we are working with a variable in k^2

When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the positive root.

That is:
\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).
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Given:
n is a positive integer.
k=5.1*10^n

Required: k = ?

Statement 1: 6,000 < k < 500,000
Since n is a positive integer, k can take the values 51, 510, 5100, 51000, 510000 etc.
Of these only one value lies in the range (6000, 500000)
Hence k = 51,000
SUFFICIENT

Statement 2: \(k^2\) = 2.601 * \(10^9\)
This will give us 2 values of k, but we are concerned with the positive value only, since k cannot be negative as per the definition k = 5.1*\(10^n\)
SUFFICIENT

Option D

could you please explain the solution of statement 2

\(k^2 = 2.601 * 10^9 = 2601*10^6 = 51^2*(10^3)^2\)

\(k= \sqrt{51^2*(10^3)^2}=51*10^3=51000\)
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Prompt analysis
K is a positive integer

Superset
The value of n could be any positive integer

Translation
In order to the value on n, we need:
1# exact value of n
2# range of k
3# any equation to find the value of n

Statement analysis
St 1: from the range, the value of n comes out to be 4. ANSWER. Hence option b,c,e eliminated
St 2: from the statement we get k = 5.1 x 10^4. Since k is positive, we can say that the value of n is 4. ANSWER

Option D.

We are given that n is a positive integer and k = 5.1*10^n and need to determine k.

Statement One Alone:

6,000 < k < 500,000

We see that if n = 4, then k = 5.1 x 10^ 4 = 51,000, which is between 6,000 and 500,000, If n = 3, then k = 5.1 x 10^3 = 5,100, which is less than 6,000 and if n = 5, then k = 5.1 x 10^5 = 510,000, which is greater than 500,000. So n must be 4 and k must be 51,000.

Statement one alone is sufficient to answer the question.

Statement Two Alone:

k^2 = 2.601 * 10^9

Since k = 5.1*10^n, and since n is a positive integer, we see that we can take the square root of both sides of the given equation and get a singular value for k. Thus, statement two is sufficient to answer the question.

Answer: D
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K = 51, 5100, 51000, 510000
1) K = 51,000
Sufficient

2) K can't be negative since n = +
Sufficient

ANSWER: D

k=5.1*10^n
(1)6,000 < k < 500,000:
The value of k in the equation will be dependent on the value of n. We will have to take the value of n such that 6,000 < k < 500,000.
For n=5, k=51000. Which also satisfies 6,000 < k < 500,000. Sufficient.

(2) k^2 = 2.601 * 10^9
Removing the square from both sides we get k=51*10^3=51000. Sufficient

D
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When i saw this question, i started solving by finding the values using calculation and realised that i spent more than 4 minutes. Your explanation and break down of solution actually helped me to understand how to solve these kind of questions within 2 minutes in exam setup. Thanks much.
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This is how I solved.

Statement 1 -
6000 < k < 500000
6000/1000 < k/1000 < 500000/1000
6 < k/500 < 500

n = 1 (No)
n = 2 (No)
n = 3 (No)
n = 4 (Yes)
n = 5 (No)
Sufficient.

Statement 2 -
k must equal \(\sqrt{2.601 x 10^9}\)

Sufficient.
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Video solution from Quant Reasoning:
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1. 6000 < k <500,000

6 *10^3 < k < 5* 10^5

Since k= 5.1 * 10^n, n has to be 4 to fit this range. Thus, A is sufficient.

2. k^2= 2.601* 10^9
= 2601 * 10^6
= (51)^2* (10^3)^2 [51 ^ 2 = 50 ^2 + 50 + 51]
k = 51 * 10^3
k = 5.1 * 10^4

Thus, n=4 and B is also sufficient. Correct answer is D.

Hi Bunuel! May I also please request to show the formula on how to solve k^2=2.601x10^9?
How did you arrive at 2601x10^6?

Please advise.

Thank you!

Regina,