GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 14 Oct 2019, 15:57

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# S96-05

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 58320

### Show Tags

16 Sep 2014, 01:50
10
00:00

Difficulty:

65% (hard)

Question Stats:

58% (02:22) correct 42% (02:07) wrong based on 105 sessions

### HideShow timer Statistics

The smallest prime factor of 899 is $$x$$. Which of the following is true of $$x$$?

A. $$1 \lt x \le 7$$
B. $$7 \lt x \le 14$$
C. $$14 \lt x \le 21$$
D. $$21 \lt x \le 28$$
E. $$28 \lt x \le 35$$

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 58320

### Show Tags

16 Sep 2014, 01:50
4
1
Official Solution:

The smallest prime factor of 899 is $$x$$. Which of the following is true of $$x$$?

A. $$1 \lt x \le 7$$
B. $$7 \lt x \le 14$$
C. $$14 \lt x \le 21$$
D. $$21 \lt x \le 28$$
E. $$28 \lt x \le 35$$

One path to the solution involves brute force. We can test primes in order of size, applying divisibility rules that we know for small numbers, such as 3. However, all the simple rules fail. This method may wind up being the quickest way, but it is laborious.

The shortcut in this problem involves wishful thinking. 899 is awfully close to a nice number: 900. The reason 900 is so nice is that it is a square: $$30^2 = 900$$. (By the way, since we know from the wording of the problem that 899 has a prime factor less than itself, at least one of the prime factors must be below the square root of 899, and at least one prime factor must be larger than the square root of 899. This square root is just under 30. This is another reason why we might think of the nearby perfect square, 900.)

So we can write $$899 = 900 - 1 = 30^2 - 1$$.

Now, ideally we would notice that we can take one step further and rewrite $$30^2 - 1$$ as $$30^2 - 1^2$$, since $$1 = 1^2$$. Why would we do this? Because now we have written 899 as a difference of squares, which we should know how to factor:
$$899 = 900 - 1 = 30^2 - 1 = 30^2 - 1^2 = (30 + 1)(30 - 1) = 31 \times 29.$$

Both 31 and 29 are prime numbers. The smallest prime factor of 899, therefore, is 29.

_________________
Intern
Status: Pursuit of Happiness
Joined: 10 Sep 2016
Posts: 29
Location: United States (IL)
Concentration: Finance, Economics
Schools: HBS '19, CBS '19
GMAT 1: 590 Q44 V27
GMAT 2: 690 Q50 V34
GPA: 3.94

### Show Tags

17 Oct 2016, 19:03
This question is a typical Manhattan question. As taught in the Advanced GMAT Quand book. Love it!

--
Need Kudos
_________________
If you find this post hepful, please press +1 Kudos
Intern
Joined: 13 Sep 2016
Posts: 2

### Show Tags

29 Apr 2017, 21:49
i think the correct answer shd be 28<x<35 and not 28<x<=35
Intern
Joined: 15 Mar 2017
Posts: 38
Location: India
GMAT 1: 720 Q50 V37
GPA: 4

### Show Tags

29 Apr 2017, 22:06
pujjwal wrote:
i think the correct answer shd be 28<x<35 and not 28<x<=35

Just look at the option trend and you will understand why is it given <=35. Anyhow both the options wont change the answer.
Hope this helps.
_________________
You give kudos, you get kudos. :D
Intern
Joined: 19 Jul 2017
Posts: 1

### Show Tags

25 Aug 2017, 17:22
I do not understand how we are able to discern that 29 was the smallest without checking if there were any numbers lower than 29. How are we sure that the only two prime factors were 29 and 31? Could someone point me to the theory behind this?
Math Expert
Joined: 02 Sep 2009
Posts: 58320

### Show Tags

26 Aug 2017, 02:23
saifsaif wrote:
I do not understand how we are able to discern that 29 was the smallest without checking if there were any numbers lower than 29. How are we sure that the only two prime factors were 29 and 31? Could someone point me to the theory behind this?

Factorise 899: 899 = 29*31. As you can see 899 does not have nay other primes but 29 and 31.
_________________
Manager
Joined: 03 Mar 2018
Posts: 206

### Show Tags

30 Apr 2018, 09:01
Hey Bunuel can you post similar questions to practice
_________________
Intern
Joined: 24 Oct 2018
Posts: 10
Location: Russian Federation
Concentration: Technology, Strategy
GPA: 3.56
WE: Information Technology (Computer Software)

### Show Tags

25 Oct 2018, 09:31
Hello,
Could you advise, how to apply this approach for 901?
Intern
Joined: 24 Oct 2018
Posts: 10
Location: Russian Federation
Concentration: Technology, Strategy
GPA: 3.56
WE: Information Technology (Computer Software)

### Show Tags

25 Oct 2018, 09:31
Hello,
Could you advise, how to apply this approach for 901?
Intern
Joined: 29 Aug 2019
Posts: 1

### Show Tags

08 Oct 2019, 19:09
In order to apply this approach you need a number to be equal to a difference of two squares. 899 equals 900 - 1. 900 is 30 squared while 1 is one squared. Therefore we could apply the identity [x^2-y^2]
901 is not a a difference of two squares, rather it is a sum of two squares so this identity does not apply. 901 is a composite number because we can see that 17 can evenly divide into it. 17 x 53 =901
Re: S96-05   [#permalink] 08 Oct 2019, 19:09
Display posts from previous: Sort by

# S96-05

Moderators: chetan2u, Bunuel