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How many prime numbers between 1 and 100 are factors of 7,15

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How many prime numbers between 1 and 100 are factors of 7150  [#permalink]

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13 Apr 2009, 01:39
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How many prime numbers between 1 and 100 are factors of 7,150?

A. One
B. Two
C. Three
D. Four
E. Five
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Joined: 02 Sep 2009
Posts: 60627
Re: How many prime numbers between 1 and 100 are factors of 7150  [#permalink]

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13 Feb 2012, 19:22
12
7
pratk wrote:
I also got 2,5,7,11 and 13 but it took me more than the two minutes to break down 143 into 11 and 13. 2,5 and 7 were the easier ones. I'm sure many will agree at in times of stress, the mind starts to play tricks. At first I stopped after 7 and thought that 143 cannot be factored further and chose C as the answer i.e. 3 prime factors but then got the other two in over the two mins.

Therefore, is there a quick way to get the prime factors of a large number?

Thanks.

Generally there is no easy way to check whether some very large number is a prime (well if it doesn't have some small primes, which are easy to check). You'll need a computer to do this.

Next, the GMAT won't give you a large number to factorize if there is no shortcut for that.

For example in our original question after you find 2, 5, and 7, just check for the next prime 11: 143/11=13.

You can also use divisibility rule for 11: if you sum every second digit and then subtract all other digits and the answer is divisible by 11, then the number is divisible by 11. So, for 143: (1+3)-4=0 --> 0 is divisible by 11 thus 143 is divisible by 11.

Or, you can notice that 143=130+13=13*10+13, so 143 must be divisible by 13.

So, as you can see there are plenty of shortcuts to get prime factorization of the numbers from the GMAT problems.

For more on divisibility rules and on verifying the primality check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.
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Re: How many prime numbers between 1 and 100?  [#permalink]

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16 Jan 2012, 12:09
4
Hate to play devil's advocate here, but the question doesn't clarify how many UNIQUE prime numbers are factors of 7150. (Which would change the answer to five). Am I missing something?
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Re: How many prime numbers between 1 and 100?  [#permalink]

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16 Jan 2012, 12:33
3
5
jgonza8 wrote:
Hate to play devil's advocate here, but the question doesn't clarify how many UNIQUE prime numbers are factors of 7150. (Which would change the answer to five). Am I missing something?

How many prime numbers between 1 and 100 are factors of 7,150?
A. One
B. Two
C. Three
D. Four
E. Five

Make prime factorization of 7,150 --> 7,150=2*5^2*11*13. So 4 prime numbers between 1 and 100 (namely 2, 5, 11, and 13) are factors of 7,150 (you shouldn't count one prime factor twice).

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Re: How many prime numbers between 1 and 100?  [#permalink]

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18 Jan 2012, 05:24
2
jgonza8 wrote:
Hate to play devil's advocate here, but the question doesn't clarify how many UNIQUE prime numbers are factors of 7150. (Which would change the answer to five). Am I missing something?

Yes, you are! Read the question again:
How many prime numbers between 1 and 100 are factors of 7,150?

The prime numbers between 1 and 100 are 2, 3, 5, 7, 11, 13, 17, 19... etc

5 appears only once between 1 and 100 so there is absolutely no confusion. Out of these 25 prime numbers, only 4 are factors of 7150: 2, 5, 11 and 13
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Re: How many prime numbers between 1 and 100 are factors of 7150  [#permalink]

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13 Feb 2012, 19:09
4
I also got 2,5,7,11 and 13 but it took me more than the two minutes to break down 143 into 11 and 13. 2,5 and 7 were the easier ones. I'm sure many will agree at in times of stress, the mind starts to play tricks. At first I stopped after 7 and thought that 143 cannot be factored further and chose C as the answer i.e. 3 prime factors but then got the other two in over the two mins.

Therefore, is there a quick way to get the prime factors of a large number?

Thanks.
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Re: How many prime numbers between 1 and 100 are factors of 7150  [#permalink]

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14 Feb 2012, 00:19
10
5
pratk wrote:
I also got 2,5,7,11 and 13 but it took me more than the two minutes to break down 143 into 11 and 13. 2,5 and 7 were the easier ones. I'm sure many will agree at in times of stress, the mind starts to play tricks. At first I stopped after 7 and thought that 143 cannot be factored further and chose C as the answer i.e. 3 prime factors but then got the other two in over the two mins.

Therefore, is there a quick way to get the prime factors of a large number?

Thanks.

Sadly, you will need to check for all prime numbers till the square root of the given number. We don't know the square root of 143 but we can approximate it. 143 is very close to 144. The square root of 144 is 12 so you will need to check for all prime numbers less than 12. If none of the prime numbers less than 12 is a factor of 143, then you can say that 143 is prime. So the question is not over till you don't check for 11 too.

So basically, you need to check for all prime factors less than $$\sqrt{n}$$ to figure out whether n is prime. If you are not sure why, check out these posts for details on factors:
http://www.veritasprep.com/blog/2010/12 ... ly-number/
http://www.veritasprep.com/blog/2010/12 ... t-squares/
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Re: How many prime numbers between 1 and 100 are factors of 7150  [#permalink]

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19 Jan 2015, 10:20
So, just to clarify this, if the question only asks for the factors we count all of the factors, no matter if there are repetitions.

Only when it specifically says that we need the "unique" factors should we diregard repeating factors.

Right?
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Re: How many prime numbers between 1 and 100 are factors of 7150  [#permalink]

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19 Jan 2015, 22:14
pacifist85 wrote:
So, just to clarify this, if the question only asks for the factors we count all of the factors, no matter if there are repetitions.

Only when it specifically says that we need the "unique" factors should we diregard repeating factors.

Right?

Not really.
Take the example of factors of 8:

How many total factors does 8 have and what are they?
They are 1, 2, 4 and 8 - a total of 4 factors

We know that 8 = 2^3 but we don't say that factors are 8 are 1, 2, 2, 2, 4 and 8.

Similarly, if we are asked - how many prime factors does 8 have? I will answer only 1 (the prime factor is 2). The number of prime factors of 8 are not 3 (not 2, 2, 2). I know of people who are not very convinced with this and hence, I assume that GMAT will insert the word "unique" to remove all doubts.

On GMAT, I would expect it to be - How many unique prime factors does 8 have?

In the original question, there is no doubt since they ask "how many prime numbers are factors of..." The set of prime numbers does not have multiple entries and hence there is no doubt that we are talking about unique prime factors only.
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Re: How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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29 Mar 2015, 15:06
Bunuel
Bunuel wrote:
How many prime numbers between 1 and 100 are factors of 7,150 ?

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

Make prime factorization of 7,150 --> $$7,150=2*5^2*11*13$$. So, 4 prime numbers between 1 and 100 (namely 2, 5, 11, and 13) are factors of 7,150.

Bunuel,

Do you have any fast tricks to understand that 7,150 is divisible by 11 and 13?

The only silly mistake I made was to miss 11 and 13 as prime factors of 7150.

Thank you!
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Re: How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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29 Mar 2015, 19:01
5
2
Hi elisabettaportioli,

The speed with which you 'prime-factor' 7150 into its 'pieces' is likely going to be influenced by the 'first' number you factor out.

Looking at 7150, you could easily start with a 2 (because 7150 is even), a 5 (because 7150 ends in a 0) or a 10 (also since it ends in a 0).

I actually started with 50, since 50 divides into 100 twice.....7100 = (71x2) fifties....

So 7150 = 142 fifties + 1 fifty =
(50)(143)

The (50) can be quickly broken down into (2)(5)(5)

Now, looking at the 143, we know that NO even numbers will divide in (since even numbers do NOT divide into odd numbers). If you know the 'rule of 3', then you know that 3 does NOT divide in. Since 3 doesn't divide in, 9 won't divide in either. 5 won't divide in for obvious reasons. Thus, we're really left with just a handful of possibilities:

1) 143 might be prime
2) 7, 11 and/or 13 might divide in

It's pretty easy to eliminate 7 as an option (it divides into 14, but not 3). Once you find that 11 divides in, you end up with the 13 by default.

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Re: How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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01 Dec 2015, 06:16
EMPOWERgmatRichC, Bunuel, VeritasPrepKarishma:
I stopped at 143 as felt it was a prime number after trying 3, 5, 7 n 9. I didnt go to 11. If i had taken that one extra step I would have gotten it right. But how far one should go? Here it is 11 but on some other questions it could be 17, 19, 23 etc. Is there a neat formula too tell all prime factors of a number? thanks
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How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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Updated on: 01 May 2016, 11:26
4
Hi NoHalfMeasures,

When you say that you tried 3, 5, 7 and 9, what actual 'work' did you do? I ask because your Quant skills are clearly strong (you have a Q48 to prove it), so I would guess that you would have figured out rather quickly that those single-digit primes wouldn't divide into 143.

Since those smaller primes don't divide in, there can't be that many left to check out (and checking EITHER 11 or 13 would have been enough work to correctly answer the question). At it's core, this is a test of 'thoroughness' - it's true that most people wouldn't think to try dividing 11, but most people can't score Q48+, so you have to decide what 'extra work' (if any) YOU'RE willing to do to be thorough, prove that your answer is correct and pick up those extra points.

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Originally posted by EMPOWERgmatRichC on 01 Dec 2015, 21:52.
Last edited by EMPOWERgmatRichC on 01 May 2016, 11:26, edited 1 time in total.
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How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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01 May 2016, 10:11
EMPOWERgmatRichC, Bunuel, VeritasPrepKarishma:

Here is my method (and I am guessing this is not the right approach, but wanted to confirm):

7150 = 7000 + 150
7000 = 7 * 1000 (so we have 7, 5 & 2)
150 = 3 * 50 (so we have 3, 5 & 2)

combining both we have 2, 3, 5 & 7 as prime numbers.

At the end, I am still getting the same number of prime numbers for 7150, albeit different values (maybe coincidence).

I will be trying this with the next similar questions I come across, till then any help....greatly appreciated.
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Re: How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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01 May 2016, 11:30
Hi colorblind,

Unfortunately, that logic is NOT correct (and it was lucky that you ended up with the correct answer). Here's a simple example that proves the logic is NOT correct.

17 has one prime factor: 17

Using the logic you described...
17 = 10+7
10 has prime factors of 2 and 5 and 7 has one prime factor: 7

However, these prime factors (re: 2, 5 and 7) are NOT the same prime factors of 17 (re: 17) and the NUMBER of prime factors is also different.

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Re: How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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02 May 2016, 07:22
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How many prime numbers between 1 and 100 are factors of 7,150 ?

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

We start by prime factoring 7,150.

7,150 = 715 x 10 = 143 x 5 x 10

At this point we must be careful. They WANT you to think that 143 is prime. Using our divisibility rules, however, we can determine that 143 is divisible by 11. A number is divisible by 11 if the sum of the odd-numbered place digits minus the sum of the even-numbered place digits is divisible by 11. We can test 143 to prove this:

1 + 3 – 4 = 4 – 4 = 0

Since zero is divisible by 11, we know that 143 is divisible by 11. We can now finish the prime factorization.

143 x 5 x 10 = 11 x 13 x 5 x 5 x 2

11 x 13 x 5^2 x 2

Thus we can see that there are 4 different prime factors in 7,150.

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How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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09 Mar 2017, 17:01
elisabettaportioli wrote:
Bunuel
Bunuel wrote:
How many prime numbers between 1 and 100 are factors of 7,150 ?

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

Make prime factorization of 7,150 --> $$7,150=2*5^2*11*13$$. So, 4 prime numbers between 1 and 100 (namely 2, 5, 11, and 13) are factors of 7,150.

Bunuel,

Do you have any fast tricks to understand that 7,150 is divisible by 11 and 13?

The only silly mistake I made was to miss 11 and 13 as prime factors of 7150.

Thank you!

One of the questions above is how to recognize easily if a number is a multiple of 11 or not.

Let $$N$$ be an integer $$a_m a_{m-1} ... a_2 a_1 a_0$$ written in 10 base notation.
Consider the difference between $$a_0 + a_2 + a_4 + ...$$, the sum of digits at even numbered positions such as 0th, 2nd, 4th, ... and $$a_1 + a_3 + a5 + ...$$, the sum of digits at odd numbered positions such 1st, 3rd, 5th, ... .
If $$( a_0 + a_2 + a_4 + ... ) - ( a_1 + a_3 + a5 + ... )$$ is a multiple of 11, then $$N$$ must be a multiple of 11.

Here, in this question, we have ( 7 + 5 ) - ( 1 + 0 ) = 12 - 1 = 11, which is a multiple of 11.
Thus 7150 must be a multiple of 11.

For 143, ( 1 + 3 ) - 4 is 0, which is also a multiple of 11.
Thus 143 is a multiple of 11.

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Re: How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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02 Jan 2018, 04:38
How many prime numbers between 1 and 100 are factors of 7,150 ?

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

Hello Everyone Can anyone suggest divisibility rule by 7, 11, and 13 ? Thanks!
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Re: How many prime numbers between 1 and 100 are factors of 7,15  [#permalink]

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02 Jan 2018, 04:50
dave13 wrote:
How many prime numbers between 1 and 100 are factors of 7,150 ?

(A) One
(B) Two
(C) Three
(D) Four
(E) Five

Hello Everyone Can anyone suggest divisibility rule by 7, 11, and 13 ? Thanks!

It's unlikely that you'll need those but you can check them here: https://gmatclub.com/forum/math-number- ... ml#p666609
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How many prime numbers between 1 and 100 are factors of 7150  [#permalink]

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27 Jun 2018, 18:35
The tag is 600-700 level. Why is the question labeled "hard" in the Official Guide 2018 Book's Onine Question Bank?
How many prime numbers between 1 and 100 are factors of 7150   [#permalink] 27 Jun 2018, 18:35

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