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is k square of an integer ?

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is k square of an integer ?

by nikhilgmat31 » Thu Jun 04, 2015 1:05 am
If k is a positive integer, is k the square of an integer?

1. k+1 is divisible by only two different numbers.

2. k is divisible by only nine different numbers.



A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.
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Source: — Data Sufficiency |
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by GMATGuruNY » Thu Jun 04, 2015 3:33 am
A better wording for the problem might be as follows:
If k is a positive integer, is k the square of an integer?

1. k+1 is has exactly two different positive divisors.

2. k has exactly 9 different positive divisors.
Statement 1: k+1 is has exactly two different positive divisors.
In other words, k+1 is PRIME.
Case 1: k+1 = 2, in which case k=1
Here, k is a perfect square.
Case 2: k+1 = 3, in which case k=2
Here, k is NOT a perfect square.
INSUFFICIENT.

Statement 2: k has exactly 9 different positive divisors.
Only a perfect square has an odd number of positive divisors.
Thus, k is a perfect square.
SUFFICIENT.

The correct answer is B.
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by nikhilgmat31 » Thu Jun 04, 2015 3:36 am
Where I can find summary of such rules.

Only a perfect square has an odd number of positive divisors.

I tried using to find number of factors using multiplying the (powers +1) of prime factors.

e.g 9 = 3^2 so 9 has 3 factors
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by Brent@GMATPrepNow » Thu Jun 04, 2015 6:57 am
For extra practice, here's a similar question with the same target question: https://www.beatthegmat.com/gmat-prep-in ... 10665.html

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by Brent@GMATPrepNow » Thu Jun 04, 2015 7:10 am
nikhilgmat31 wrote:If k is a positive integer, is k the square of an integer?

1. k+1 is divisible by only two different numbers.
2. k is divisible by only nine different numbers.
Notice that Mitch reworded the question so that it's more GMAT-like.
The GMAT almost always restricts factors to POSITIVE integers. In the original question, such a restriction is not made, which means that we need to consider NEGATIVE factors as well (which would ruin the question)

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by DavidG@VeritasPrep » Thu Jun 04, 2015 7:24 am
I tried using to find number of factors using multiplying the (powers +1) of prime factors.
You had the right idea. If you'd taken 2^8, for example, you can see that there would be 8+1 = 9 factors. And 2^8 is the same as (2^4)^2, so it's clearly a perfect square.

Similarly 3^8 will also have 9 factors, and 3^8 = (3^4)^2, so this is also a perfect square. Note: any prime base raised to an even exponent will be a perfect square, because squaring a number is the equivalent of multiplying all of its exponent values by 2.

Put another way, any positive number, when expressed as a product of its prime bases, will be a perfect square if each of those prime bases is raised to an even exponent. So (2^10) * (3^12) is a perfect square, as '10' and '12' are both even. But (2^9 * 3^12) is not a perfect square, as '9' is not even.

The point: rather than memorizing a list of rules, which can offer some benefits, it's often better to really understand the mechanics and logic of those rules, so they can be applied to other scenarios.
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