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. \(28 \lt x \le 35\)
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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.


Answer: E
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This question is a typical Manhattan question. As taught in the Advanced GMAT Quand book. Love it!

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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.

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.
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Hey Bunuel can you post similar questions to practice
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Hello,
Could you advise, how to apply this approach for 901?

Hello,
Could you advise, how to apply this approach for 901?

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

899 = 900-1
\(=30^2-1^2\)
=(30+1)(30-1)
=31*29, both are primes.