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

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Math Expert
Joined: 02 Sep 2009
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The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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11 Nov 2014, 09:06
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Tough and Tricky questions: Number Properties.

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. $$28 \lt x \le 35$$

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Re: The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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11 Nov 2014, 10:23
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Answer choice E. I was not sure how to attack this one and am sure there is a much easier way. If your long division skills are up to par, then it shouldn't take more than 2 minutes to answer. Each answer choice offers only a couple of possibilities for x so I just divided 899 by each possibility until one of them worked.

A. can be eliminated because 899 is not divisible by 2, 3, or 5
B. can be eliminated because 899 is not divisible by 11 or 13
C. can be eliminated because 899 is not divisible by 17 or 19
D. can be eliminated because 899 is not divisible by 23
E. this tells us x is either 29 or 31 and 899 is divisible by both (29x31=899)

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Re: The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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11 Nov 2014, 12:24
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Bunuel wrote:

Tough and Tricky questions: Number Properties.

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. $$28 \lt x \le 35$$

Kudos for a correct solution.

Not sure if this would work every time, but here is how I approached this, and it worked:

Since the GMAT doesn't typically expect you to do a lot of heavy calculations, I tried to find something special about the number. I realized that 899 is very close to 900, which is a perfect square of 30.

Therefore:
899 = 30^2 - 1
899 = 30^2 - (1^2)
899 = (30-1)(30+1)
899 = (29)(31)

Since both 29 and 31 are prime numbers, I can't factor them any further. The correct answer is E.
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Re: The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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11 Nov 2014, 12:30
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Bunuel wrote:

Tough and Tricky questions: Number Properties.

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. $$28 \lt x \le 35$$

Kudos for a correct solution.

Note that 899 <900 and $$\sqrt{900}$$=30...so prime factor is below 30...

Note that it is always better to start with C option as we can easily rule 2,3,5 and 7...

In Option D between 21 and 28 there is only 1 prime and we can find whether 23 is that factor or not quickly...It is not because $$899/23\neq {Integer}$$

In option E we have 29,31, as prime but 31 is ruled out...and 899/29=Integer so ans is E..
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Re: The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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11 Nov 2014, 19:54
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899 = 900 - 1

$$= 30^2 - 1^2$$

= (30+1)(30-1)

= 31 * 29

29 is the smallest factor

One more approach

899 = 1000 - 100 - 1

$$= 10^3 - 10^2 - 1$$

$$= 10^2(10-1) - 1$$

$$= 10^2 * 3^2 - 1$$

$$= (10*3)^2 - 1^2$$

= (30+1)(30-1)

= 31 * 29
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Posts: 44413
Re: The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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12 Nov 2014, 04:13
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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.

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The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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19 Jul 2015, 10:28
I've noticed that for many "medium" and "hard" level questions, there are tricks and ways to look at it that simplify the answer. If 899 is really close to 900, there's probably a reason for it and 900 can be used in some way or another, as the GMAT won't ask you a question that can't be done within a reasonable amount of time (e.g. a question won't require a math genius to do a problem by purely chugging through it in 10 minutes - no Good Will Hunting-hard questions here). It's daunting to look at, at first, but it's simpler if you keep that in mind - it's a common theme on GMAT questions and a good strategy to utilize. Of course, this sort of question is much easier if you already know how to approach this kind of question.
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Re: The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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23 Aug 2016, 06:41
I solved it by two methods
First method => Brute Force
Here we have to check everything and would take about 3 minutes (maybe 4 for someone like me)
Best way is to look for a technique so that we can prime factorize
Here 899 = 900-1=> 30^2-1^2=> 31*29 => BOOM ...!!!! 29 is the smallest prime factor
SMASH that E

First 26 primes =>
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
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Re: The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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29 Nov 2016, 15:43
899 = 900-1 (fancy little way to utilize what we know about (x2-1))
......= (30+1)(30-1) = (31)(29) --> BOTH PRIMES

Thus E is the correct answer.
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Re: The smallest prime factor of 899 is x. Which of the following is true [#permalink]

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02 Dec 2017, 13:32
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Re: The smallest prime factor of 899 is x. Which of the following is true   [#permalink] 02 Dec 2017, 13:32
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