How many integers between 15 and 35, inclusive, have exactly 3 prime

Question

How many integers between 15 and 35, inclusive, have exactly 3 prime factors, not necessarily unique, in their prime factorization? (e.g., the prime factorization of 8 is 2, 2, and 2, so the integer 8 would have 3 such prime factors)

A) 2
B) 3
C) 4
D) 5
E) 6

Source: Self-made question.

Show Answer

Official Answer

D

A) 2
B) 3
C) 4
D) 5
E) 6

Source : Self-made question.

Valeron · Answer

D) 5

18 = 2*3*3
20 = 2*2*5
27 = 3*3*3
28 = 2*2*7
30 = 2*3*5

Posted from my mobile device

MathRevolution · Answer

As per the example you gave, we have 27 = 3 * 3 * 3

Other integers:

=> 18 = 2 * 3 * 3

=> 20 = 2 * 2 * 5

=> 28 = 2 * 2 * 7

=> 30 = 2 * 3 * 5

Total = 5

Answer D

AndrewN · Answer

Hello, everyone. I was interested in creating a question that tasked the solver to think of numbers in different ways. I was curious whether the example would make the problem too easy or, at the very least, expedite the process of solving the question. To speak to the latter point, if we know that 8 already has 3 prime factors, then we also know that no multiple of 8 between 15 and 35, inclusive, can also have 3 prime factors (since 8 * 2, 8 * 3, and 8 * 4 would all introduce new prime factors). We can decide whether to eliminate numbers by counting up in succession or by singling out different types of numbers one by one. For instance, we could get rid of all the primes after eliminating all the multiples of 8:

Step 1: 15  16  17 18 19 20 21 22 23  24  25 26 27 28 29 30 31  32  33 34 35

Step 2: 15  17  18  19  20 21 22  23  25 26 27 28  29  30  31  33 34 35

What then? I will admit to combing through each integer one by one.

Step 3 (or step 1, depending on your method):

15 = 3 * 5  X 
18 = 2 * (3 * 3)  √ 
20 = (2 * 2) * 5  √ 
21 = 3 * 7  X 
22 = 2 * 11  X 
25 = 5 * 5  X 
26 = 2 * 13  X 
27 = (3 * 3 * 3)  √ 
28 = (2 * 2) * 7  √ 
30 = 2 * 3 * 5  √ 
33 = 3 * 11  X 
34 = 2 * 17  X 
35 = 5 * 7  X

Of course, there are   five   integers with the prime factorization that the question asks about, so   the answer is (D)  , 5. But notice how different the qualifying integers are from one another:

18, 20, and 28 all pair a prime number with a perfect square   27 is the only perfect cube   30 is the only integer that has three  unique  prime factors

Therein lies my interest. Someone seeking out a pattern and sticking strictly to it might miss 27 or 30, and the successive qualifying integers 27 and 28 might make a person doubt whether he or she had caught all the other qualifying integers beforehand. In any case, I had fun drumming up the question, and I hope it was fun to solve, too.

As always, good luck with your studies.

- Andrew

dcn1102 · Answer

Can someone tell me why 12 is not a part of it

12 = 2x2x3

Shikhar22 · Answer

Because the range asked is from 15 to 35?

Posted from my mobile device

CrackverbalGMAT · Answer

These would be the integers b/w 15-35with 3 prime factors:-
18 , 20,27,28,30

=>18 = 2*3*3

=>20 = 2*2*5

=>27 = 3*3*3

=>28 = 2*2*7

=>30 = 2*3*5  (option d)

Devmitra Sen
GMAT SME

elzein777 · Answer

What about 24 = 2x3x4

Posted from my mobile device

AndrewN · Answer

The problem with 24,  elzein777 , is that 4 can be factored into 2 * 2, meaning that prime factorization of 24 is 2 * 3 * (2 * 2). There are four prime factors when the question asks for three. If you need further explanation for any of the numbers within the range specified in the question, see my earlier post above.

Good luck with your studies.

- Andrew

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

How many integers between 15 and 35, inclusive, have exactly 3 prime

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

How many integers between 15 and 35, inclusive, have exactly 3 prime

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards