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

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

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17 Sep 2010, 03:14
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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

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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, 03:29
43
55
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

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:
http://gmatclub.com/forum/everything-ab ... 85592.html

Hope it helps.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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15 Feb 2011, 20:36
42
1
10
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
##### General Discussion
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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20 Oct 2010, 04:09
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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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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20 Oct 2010, 04:14
1
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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If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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14 Feb 2011, 06:34
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Manager
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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15 Feb 2011, 20:36
7
2
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, what is  [#permalink]

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15 Nov 2012, 12:10
Bunuel wrote:
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.

How did you know that 2 factors and 5 factors in N are same?
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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16 Nov 2012, 03:32
1
Amateur wrote:
Bunuel wrote:
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.

How did you know that 2 factors and 5 factors in N are same?

No, that's not what I'm saying (see the red part). The power of 2 in N is at least as high as the power of 5 in N.

We are told that N=3*6*9*12*15*18*21*...*90*93*96*99 --> as you can observe, the power of 2 in N will be higher than the power of 5 (there are more even numbers than multiples of 5).

Hope it's clear.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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Updated on: 23 Jan 2015, 09:04
3
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.

Originally posted by Amateur on 16 Nov 2012, 07:58.
Last edited by Amateur on 23 Jan 2015, 09:04, edited 2 times in total.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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30 Dec 2012, 19:21
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
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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31 Dec 2012, 03:34
7
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.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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04 Jun 2013, 05:48
18
2
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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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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17 Aug 2013, 00:58
1
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

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

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22 Aug 2013, 03:44
1
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.
"
once in 15;
once in 30;
once in 45;
once in 60;
twice in 75 (5*5*3);
once in 90;
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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22 Aug 2013, 03:53
1
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.
"
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

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

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25 Sep 2013, 12:18
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?
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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25 Sep 2013, 13:34
1
1
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!
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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25 Sep 2013, 21:23
1
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.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is  [#permalink]

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25 Sep 2013, 23:35
2
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.\

Re: If N is the product of all multiples of 3 between 1 and 100, what is   [#permalink] 25 Sep 2013, 23:35

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