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Question on Perfect Square 5

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by richachampion » Mon Oct 17, 2016 3:07 am
The number N is not a perfect square. There are 8 factors of N between 1 and N^0.5
How many total factors does N have ?
A) 16
B) 18
C) 19
D) 20
E) 22
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by richachampion » Mon Oct 17, 2016 3:08 am
OA: B
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by Brent@GMATPrepNow » Mon Oct 17, 2016 6:47 am
I have added "INTEGER" to the question to make it more GMAT-like.
richachampion wrote:The INTEGER N is not a perfect square. There are 8 factors of N between 1 and N^0.5 How many total factors does N have ?

A) 16
B) 18
C) 19
D) 20
E) 22
Note: N^(0.5) = √N

If integer K is not a perfect square then HALF of the positive factors of K will be LESS THAN √K and HALF of the positive factors of K will be GREATER THAN √K.

Example: the factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24
The square root of 24 equals 4.something. So, as we can see, 4 of the factors are less than 4.something, and 4 of the factors are greater than 4.something

In this question, we're told that N is not a perfect square and there are 8 factors BETWEEN 1 and √N. So, we are not including 1 as a factor. Since 1 is a factor of all integers, we can conclude that there are 9 factors LESS THAN √N, which means that are 9 factors GREATER THAN √N.

So, the total number of factors = 9 + 9 = 18
Answer: A

RELATED VIDEOS
- Introduction to exponents: https://www.gmatprepnow.com/module/gmat ... video/1021
- Squares of integers: https://www.gmatprepnow.com/module/gmat ... /video/829
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by richachampion » Mon Oct 17, 2016 10:40 am
Brent@GMATPrepNow wrote:I have added "INTEGER" to the question to make it more GMAT-like.
Thanks for the amendment sir and thanks for making it in a structured and correct format.
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by Matt@VeritasPrep » Fri Oct 28, 2016 12:35 am
Another way of thinking about this: if N is not a perfect square, its factors can be broken into pairs whose product is N. For instance, suppose N = 30. Then we have

30 = 1 * 30
30 = 2 * 15
30 = 3 * 10
30 = 5 * 6

We've got THREE factors between 1 and √30, so we've got 3 pairs of factors, plus 1 and 30 itself.

From there, we can generalize. If N is a positive integer that is not a perfect square, and N has x factors between 1 and √N, then N has 2*x + 1 + 1 factors: the x pairs, the number 1, and x itself.
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