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# If N is the product of all multiples of 3 between 1 and 100

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If N is the product of all multiples of 3 between 1 and 100 [#permalink]

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17 Sep 2010, 02:14
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If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which $$\frac{N}{10^m}$$ is an integer?

A. 3
B. 6
C. 7
D. 8
E. 10

How do you solve these sort of questions quickly
Thanks
[Reveal] Spoiler: OA

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17 Sep 2010, 02:29
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rafi wrote:
If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which $$\frac{N}{10^m}$$ is an integer?

a. 3
b. 6
c. 7
d. 8
e. 10

How do you solve these sort of questions quickly
Thanks

We should determine # of trailing zeros of N=3*6*9*12*15*...*99 (a sequence of 0's of a number, after which no other digits follow).

Since there are at least as many factors 2 in N as factors of 5, then we should count the number of factors of 5 in N and this will be equivalent to the number of factors 10, each of which gives one more trailing zero.

Factors of 5 in N:
once in 15;
once in 30;
once in 45;
once in 60;
twice in 75 (5*5*3);
once in 90;

1+1+1+1+2+1=7 --> N has 7 trailing zeros, so greatest integer $$m$$ for which $$\frac{N}{10^m}$$ is an integer is 7.

Check this for more:

Hope it helps.
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15 Feb 2011, 19:36
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N = The product of the sequence of 3*6*9*12....*99

N therefore is also equal to 3* (1*2*3*.....*33)

Therefore N = 3* 33!

From here we want to find the exponent number of prime factors, specifically the factors of 10.

10 = 5*2 so we want to find which factors is the restrictive factor

We can ignore the 3, since a factor that is not divisible by 5 or 2 is still not divisible if that number is multiplied by 3.

Therefore:

33/ 2 + 33/4 + 33/8 = 16+8+4 = 28

33/ 5 + 33/25 = 6 + 1 = 7

5 is the restrictive factor.

Here is a similar problem: number-properties-from-gmatprep-84770.html
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Re: If N is the product of all multiples of 3 between 1 and 100 [#permalink]

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04 Jun 2013, 04:48
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rafi wrote:
If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which $$\frac{N}{10^m}$$ is an integer?

A. 3
B. 6
C. 7
D. 8
E. 10

How do you solve these sort of questions quickly
Thanks

Responding to a pm:

First, check out this post. It is an application of a concept that discusses the maximum power of a number in a factorial. This post discusses how and why we find the maximum power.
http://www.veritasprep.com/blog/2011/06 ... actorials/

Once you are done, note that this question can be easily broken down into the factorial form.

$$3*6*9*...*99 = 3^{33} * (1*2*3*4*...*32*33) = 3^{33} * 33!$$

We need to find the number of 5s in 33! because you need a 2 and a 5 to make a 10. The number of 5s will certainly be fewer than the number of 2s.

33/5 = 6
6/5 = 1

So you will have a total of 6+1 = 7 5s and hence can make 7 10s.
So maximum power of 10 must be 7.

Note that we ignore $$3^{33}$$ because it has no 5s in it.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 07 Jun 2010 Posts: 86 Followers: 1 Kudos [?]: 32 [5] , given: 0 Re: Product of sequence [#permalink] ### Show Tags 15 Feb 2011, 19:36 5 This post received KUDOS 1 This post was BOOKMARKED N = The product of the sequence of 3*6*9*12....*99 N therefore is also equal to 3* (1*2*3*.....*33) Therefore N = 3* 33! From here we want to find the exponent number of prime factors, specifically the factors of 10. 10 = 5*2 so we want to find which factors is the restrictive factor We can ignore the 3, since a factor that is not divisible by 5 or 2 is still not divisible if that number is multiplied by 3. Therefore: 33/ 2 + 33/4 + 33/8 = 16+8+4 = 28 33/ 5 + 33/25 = 6 + 1 = 7 5 is the restrictive factor. Here is a similar problem: number-properties-from-gmatprep-84770.html Manager Joined: 05 Nov 2012 Posts: 171 Followers: 1 Kudos [?]: 33 [3] , given: 57 If N is the product of all multiples of 3 between 1 and 100 [#permalink] ### Show Tags 16 Nov 2012, 06:58 3 This post received KUDOS I did it in a different way..... since it is multiplication of all 3 multiples.... 3*6*9*..... *99=(3^33)(1*2*3*4*5*......33)=(3^33)*33! (3 power 33 because a 3 can be extracted from each number inside) (3^33) doesn't have any multiples between 1-9 which can contribute a 0..... so number of trailing 0's should be number of trailing 0's of 33! which is 7. So C is the answer... we don't need to count 5's and 2's and complicate things in this case! Let me know if you think this approach of mine has loop holes. Last edited by Amateur on 23 Jan 2015, 08:04, edited 2 times in total. Math Expert Joined: 02 Sep 2009 Posts: 37144 Followers: 7272 Kudos [?]: 96805 [3] , given: 10786 Re: If N is the product of all multiples of 3 between 1 and 100 [#permalink] ### Show Tags 31 Dec 2012, 02:34 3 This post received KUDOS Expert's post lunar255 wrote: I am not convinced by the answer of Bunuel, so I used excel to calculate the product. The answer is 48,271,088,561,614,000,000,000,000,000,000,000,000,000,000,000,000,000, which means the maximum of m will be 39. This is not a good question 1. There is nothing wrong with the question. 2. Solution is correct, answer is C. 3. Excel rounds big numbers. Actual result is 48,271,088,561,613,960,642,858,365,853,327,381,832,862,269,440,000,000. _________________ Magoosh GMAT Instructor Joined: 28 Dec 2011 Posts: 3857 Followers: 1332 Kudos [?]: 6132 [2] , given: 73 If N is the product of all multiples of 3 between 1 and 100, what is t [#permalink] ### Show Tags 18 May 2015, 15:46 2 This post received KUDOS Expert's post reto wrote: If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which $$\frac{N}{10^m}$$is an integer? A. 3 B. 6 C. 7 D. 8 E. 10 Dear Reto, My friend, before you post anything else, please familiarize yourself with the protocols. This question has been posted many times before, for example, here: if-n-is-the-product-of-all-multiples-of-3-between-1-and-101187.html where there's already a long discussion. Always search for a question before you start a new thread from scratch. Presumably, Bunuel, the math genius moderator, will merge this post into one of the larger previous posts on the same topic. If you have any questions that are not already answered there, you are more than welcome to ask me. Best of luck, Mike _________________ Mike McGarry Magoosh Test Prep Intern Joined: 03 Apr 2012 Posts: 27 Followers: 0 Kudos [?]: 12 [1] , given: 10 Re: If N is the product of all multiples of 3 between 1 and 100 [#permalink] ### Show Tags 16 Aug 2013, 23:58 1 This post received KUDOS I think the easiest way to do it is to count the number of 5's from 1 to 33. 3^ 33 ( 1 x 2x 3...... 33) 5 factors 5 - 5x1 10- 5x2 15- 5x3 20 - 5x4 25 - 5x5 30 - 5x6 Therefore the answer is 7. Math Expert Joined: 02 Sep 2009 Posts: 37144 Followers: 7272 Kudos [?]: 96805 [1] , given: 10786 Re: If N is the product of all multiples of 3 between 1 and 100 [#permalink] ### Show Tags 22 Aug 2013, 02:53 1 This post received KUDOS Expert's post mumbijoh wrote: Dear Bunuel I came across this question and i really do not understand it.I read the "Everything about factorial " link but i cant seem to apply what i have read there to this question. How did you come up with this?Please help " once in 15; once in 30; once in 45; once in 60; twice in 75 (5*5*3); once in 90; 15=5*3 30=5*6 45=5*9 60=5*12 75=5^2*3 90=5*18 Similar questions to practice: if-n-is-the-greatest-positive-integer-for-which-2n-is-a-fact-144694.html what-is-the-largest-power-of-3-contained-in-103525.html if-n-is-the-product-of-all-positive-integers-less-than-103218.html if-n-is-the-product-of-integers-from-1-to-20-inclusive-106289.html if-n-is-the-product-of-all-multiples-of-3-between-1-and-101187.html if-p-is-the-product-of-integers-from-1-to-30-inclusive-137721.html what-is-the-greatest-value-of-m-such-that-4-m-is-a-factor-of-105746.html if-6-y-is-a-factor-of-10-2-what-is-the-greatest-possible-129353.html if-m-is-the-product-of-all-integers-from-1-to-40-inclusive-108971.html if-p-is-a-natural-number-and-p-ends-with-y-trailing-zeros-108251.html if-73-has-16-zeroes-at-the-end-how-many-zeroes-will-147353.html find-the-number-of-trailing-zeros-in-the-expansion-of-108249.html how-many-zeros-are-the-end-of-142479.html how-many-zeros-does-100-end-with-100599.html find-the-number-of-trailing-zeros-in-the-product-of-108248.html if-60-is-written-out-as-an-integer-with-how-many-consecuti-97597.html if-n-is-a-positive-integer-and-10-n-is-a-factor-of-m-what-153375.html if-d-is-a-positive-integer-and-f-is-the-product-of-the-first-126692.html Hope it helps. _________________ Math Expert Joined: 02 Sep 2009 Posts: 37144 Followers: 7272 Kudos [?]: 96805 [1] , given: 10786 Re: Number properties task, please, help! [#permalink] ### Show Tags 25 Aug 2013, 10:29 1 This post received KUDOS Expert's post pavan2185 wrote: Bunuel wrote: Can you please tell what do you find most challenging in them? Thank you. Check other similar questions here: if-n-is-the-product-of-all-multiples-of-3-between-1-and-101187-20.html#p1259389 I understand the basic concept you explained in the mathbook by Gmatclub and various explanations you have given,but I am finding it difficult to apply on hard questions that involve multiple factorilas and questions that do not specifically give any fcatorial but give a complex product of numbers. In that case, I must say that practice should help. _________________ MBA Section Director Status: Back to work... Affiliations: GMAT Club Joined: 21 Feb 2012 Posts: 4044 Location: India City: Pune GMAT 1: 680 Q49 V34 GPA: 3.4 WE: Business Development (Manufacturing) Followers: 408 Kudos [?]: 2969 [1] , given: 2205 Re: If N is the product of all multiples of 3 between 1 and 100 [#permalink] ### Show Tags 25 Sep 2013, 12:34 1 This post received KUDOS Expert's post TAL010 wrote: Finding the powers of a prime number p, in the n! The formula is: Example: What is the power of 2 in 25!? ^^ Taken from the GMAT Club book...what is the logic behind this question? What are they really asking? It means calculating number of instances of P in n! Consider the simple example ---> what is the power of 3 in 10! We can find four instances of three in 10! -----> 1 * 2 * 3 * 4 * 5 * (2*3) * 7 * 8 * (3*3) * 10 You can see above we can get four 3s in the expression. Calculating the number of instances in this way could be tedious in the long expressions. but there is a simple formula to calculate the powers of a particular prime. the powers of Prime P in n! can be given by $$\frac{n}{p} + \frac{n}{p^2} + \frac{n}{p^3} + .................$$ till the denominator equal to or less than the numerator. what is the power of 3 in 10! ------> $$\frac{10}{3} + \frac{10}{3^2} = 3 + 1 = 4$$ Analyze how the process works........ We first divided 10 by 1st power of 3 i.e. by 3^1 in order to get all red 3s Later we divided 10 by 2nd power of 3 i.e. by 3^2 in order to get the leftover 3 (blue) we can continue in this way by increasing power of P as long as it does not greater than n Back to the original question.............. What is the power of 2 in 25!? ---------> 25/2 + 25/4 + 25/8 + 25/16 = 12 + 6 + 3 + 1 = 22 Hope that helps! _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7191 Location: Pune, India Followers: 2173 Kudos [?]: 14058 [1] , given: 222 Re: If N is the product of all multiples of 3 between 1 and 100 [#permalink] ### Show Tags 25 Sep 2013, 20:23 1 This post received KUDOS Expert's post TAL010 wrote: Finding the powers of a prime number p, in the n! The formula is: Example: What is the power of 2 in 25!? ^^ Taken from the GMAT Club book...what is the logic behind this question? What are they really asking? Check out this post: http://www.veritasprep.com/blog/2011/06 ... actorials/ It answers this question in detail explaining the logic behind it. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: If N is the product of all multiples of 3 between 1 and 100 [#permalink]

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25 Sep 2013, 22:35
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We know that for a number to be divisible by 10 must have at least one zero. Let's break the 10 into its prime factors, ie. 5 and 2. Now, we need to find pairs of 2 and 5 in the numerator. Here, 5 is our limiting factor, as it appears less than 2 does. therefore two cont the number of 5s, we must count the 5s in all multiples of 3 between 1 and 100.

15= One 5
30= One 5
45= One 5
60= One 5
75 = Two 5s (5 x 5 x3=75)
90= One 5.\

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Re: If N is the product of all multiples of 3 between 1 and 100 [#permalink]

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23 May 2015, 02:24
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reto wrote:
VeritasPrepKarishma wrote:
rafi wrote:
If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which $$\frac{N}{10^m}$$ is an integer?

A. 3
B. 6
C. 7
D. 8
E. 10

How do you solve these sort of questions quickly
Thanks

Responding to a pm:

First, check out this post. It is an application of a concept that discusses the maximum power of a number in a factorial. This post discusses how and why we find the maximum power.
http://www.veritasprep.com/blog/2011/06 ... actorials/

Once you are done, note that this question can be easily broken down into the factorial form.

$$3*6*9*...*99 = 3^{33} * (1*2*3*4*...*32*33) = 3^{33} * 33!$$

We need to find the number of 5s in 33! because you need a 2 and a 5 to make a 10. The number of 5s will certainly be fewer than the number of 2s.

33/5 = 6
6/5 = 1

So you will have a total of 6+1 = 7 5s and hence can make 7 10s.
So maximum power of 10 must be 7.

Note that we ignore $$3^{33}$$ because it has no 5s in it.

Dear Karishma

Could you explain step by step how to arrive at $$3^{33}*33!$$ It's logical for me that we have to illustrate the product of all multiples of 3 between 1-100. The following is however not quite clear for me:

1. Did you count all the multiples of 3 between 1 and 100 "manually" or is there a smart way?
2. Why do you multiply by 33! ?

Could you help me here?
Thank you!

You don't have to count the multiples of 3. Just look at the pattern.

Multiples of 3:

3 * 6 * 9 * 12 * ... * 96 * 99

3 = 3*1
6 = 3*2
9 = 3*3
...
96 = 3*32
99 = 3*33

So in all, we have 33 multiples of 3.

(3*1) * (3*2) * (3*3) * (3*4) * ... * (3*32) * (3*33)

Now from each term, separate out the 3 and put all 3s together in the front. You have 33 terms so you will get 33 3s. Also you will be left with all second terms 1, 2, 3, 4 etc

= (3*3*3..*3) * (1 * 2 * 3 * 4 * ... * 32 * 33)

= 3^(33) * (1 * 2 * 3 * 4 * ... * 32 * 33)

But 33! = (1 * 2 * 3 * 4 * ... * 32 * 33)

So you get 3^(33) * 33!
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20 Oct 2010, 03:09
Bunuel, is it necessary to count the number of trailing zeros? I have solved the problem by counting the number of 5's in N.
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20 Oct 2010, 03:14
nonameee wrote:
Bunuel, is it necessary to count the number of trailing zeros? I have solved the problem by counting the number of 5's in N.

It's basically the same. Since there are at least as many factors 2 as factors of 5 in N, then finding the number of factors of 5 in N would be equivalent to the number of factors 10, each of which gives one more trailing zero.
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14 Feb 2011, 05:11
I am only able to get ans B (6) and not the OA: 7(C).

Allow me to share my take on this,

Since N which is the product of all the multiples of 3 between 1 and 100 i.e.
N = 3X6X9X12X.....99,

for N to be divided by 10 and remain an integer, I need to find out the number of factors with "0" in the ones digit

N contains factors 30,60 and 90 so m will be at least 3 since (30X60X90)/1000 is an integer

and since 10 = 5X2, any multiple of 3 with a "5" as a ones digit when multiplied by an even number will yield a number with "0" in the ones digit.

So 15,45 and 75 (all multiples of 3) come to mind. Since there are plenty of even number factors in N, I get another 3 for the value of m

So m = 3+3 = 6. I dont get how m can be 7 though.. so am i missing something?
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14 Feb 2011, 05:17
IrinaTyan wrote:
If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which is N/(10^m) is an integer?

1. 3
2. 6
3. 7
4. 8
5. 10

Add up the terms that can lead to a zero that are multiples of 3

30
60
90
15*12
45*42
75*72

cant think of the 7th and gota run to work, but that is how you do it!
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14 Feb 2011, 05:34
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