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

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rakeshd347: Master | Next Rank: 500 Posts; Posts: 391; Joined: Sat Mar 02, 2013 5:13 amFollowed by: 4 members; Thanked: 50 times

by rakeshd347 » Wed Oct 02, 2013 7:25 pm

If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which N/10^m is an integer?

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

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Wed Oct 02, 2013 8:41 pm

Is the Answer [spoiler]{C}[/spoiler]

Multiple of 3 from 1 to 100 means
3(1.2.3.4....33) ==> 3.33!

So, the question is basically asking for MAXIMUM power of 10 in N

33/5 = 6
6/5 = 1

6+1 = 7

R A H U L

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mevicks: Master | Next Rank: 500 Posts; Posts: 269; Joined: Thu Sep 19, 2013 12:46 amFollowed by: 7 members; Thanked: 94 times

by mevicks » Wed Oct 02, 2013 8:46 pm

N = 3 x 6 x 9 x ... 99

N when divided by 10^m should yield an integer result.

We just need to get hold of all the multiples of 10 & 5
5 --> Because 5*2 = 10 so each 5 can form a pair with an even number (plenty in the list of multiples of 3)

Special mention of 75. 75 has two 5s in it (5 * 5 * 2) so it can cancel out 1 additional 10 when grouped with a even number. Thus we can form 7 10s out of the denominator 6 + 1 ( additional 5 from 75)
Answer : [spoiler]m = 7[/spoiler]

Regards,
Vivek

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ganeshrkamath: Master | Next Rank: 500 Posts; Posts: 283; Joined: Sun Jun 23, 2013 11:56 pmFollowed by: 26 members; Location: Bangalore, India; Thanked: 97 times; GMAT Score:750

by ganeshrkamath » Wed Oct 02, 2013 11:48 pm

rakeshd347 wrote:If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which N/10^m is an integer?

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

We have to find the multiples of 3 which are also multiple of 2 or 5 or both:
3*(2,4,5,6,8,10,12,14,15,16,18,20,22,24,25,26,28,30,32)
Power of 5 = 5^(1 + 1 + 1 + 1 + 2 + 1) = 5^7
So we can have a maximum of 7th power of 10.

Choose C

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ani781: Master | Next Rank: 500 Posts; Posts: 141; Joined: Fri Jun 24, 2011 4:35 amFollowed by: 1 member; Location: Edison; Thanked: 12 times

by ani781 » Thu Oct 03, 2013 5:11 am

@Rahul
Can you pls elaborate on the solution that you provided. This is not clear to me

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Thu Oct 03, 2013 5:28 am

ani781 wrote:@Rahul
Can you pls elaborate on the solution that you provided. This is not clear to me

According to question, "N is the product of all multiples of 3 between 1 and 100,"

STEP 1:
we can re-write this as

3x6x9x12x15x18x....99

So, 3 (1x2x3x4x5.....33) --> 3.33!

STEP 2:
We need greatest integer "m" for N/10^m

that is, we can we-write this question as, we need the find the maximum power of "10" in "N" (3.33!)

IMP: There's a very simple and helpful logic, whenever you need to find the power of any number in any factorial.. divide the number(removing factorial) until you cannot divide further; For power of 10, divide by "5"

SO,
33/5 = 6 [we are using "33" and not "33!"]
6/5 = 1 [re-divide]
1/5 = 0 [no further division possible; terminate]

So, Maximum power = 6+1 = 7

I hope the solution is now much more understandable

Recently, many similar questions were posted, you may refer and try to apply this method on those question as well..

R A H U L

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by [email protected] » Thu Oct 03, 2013 10:44 am

Hi rakeshd347,

This question essentially asks us to find all of the "10s" that can found (or multiplied together from a 2 and 5) in the string of multiples of 3 (from 3 to 99).

While there are a number of ways to solve this problem, here's how I did it:

Find all the 5s and 10s:

15 = has a 5
30 = has a 10
45 = has a 5
60 = has a 10
75 = has TWO 5s
90 = has a 10

Since all of the even multiples of 3 have a 2 within it (eg 6 = 3x2, 12 = 3x2x2, etc.), we can take those 2s and multiply all of the 5s in the above table to form 10s.

Since we have three 10s and four 5s (which we'll multiply by 2), we'll end up with SEVEN 10s.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Oct 03, 2013 11:33 am

rakeshd347 wrote:If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which N/10^m is an integer?

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

N = 3*6*9*...*93*96*99 = (3Â³Â³)(1*2*3*...*31*32*33) = 3Â³Â³ * 33!.
10^m = the number of 10's that can divide into N.

We need to count how many 10's can divide into 33!.
Since 10=2*5, EVERY COMBINATION OF 2*5 contained within the prime-factorization of 33! will provide a 10 than can divide into N.
The prime-factorization of 33! will be composed of FAR MORE 2'S than 5's.
Thus, to determine HOW MANY 10'S can divide into 33!, we must count THE NUMBER OF 5'S contained within the prime-factorization of 33!.

To determine the number of 5's, count how many times EACH POWER OF 5 can divide into 33.

Every multiple of 5 that can divide into 33 provides at least one 5:
33/5Â¹ = 6.

Every multiple of 5Â² that can divide into 33 adds a second 5:
33/5Â² = 33/25 = 1.

Thus, the greatest number of 10's than can divide into N = 6+1 = 7.

The correct answer is C.

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https://www.beatthegmat.com/product-of-t ... 94247.html
https://www.beatthegmat.com/if-k-is-the- ... 49798.html

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ani781: Master | Next Rank: 500 Posts; Posts: 141; Joined: Fri Jun 24, 2011 4:35 amFollowed by: 1 member; Location: Edison; Thanked: 12 times

by ani781 » Thu Oct 03, 2013 1:37 pm

Thanks Rich, you explanation is crystal clear.
I dont know why , but maybe because my medium of study was non-english, I could not at all understand the problem... untill I saw your solution.

Thanks !

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by Scott@TargetTestPrep » Fri Nov 17, 2017 1:37 pm

rakeshd347 wrote:If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which N/10^m is an integer?

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

We can rewrite our question:

What is the greatest integer m for which N/(2^m x 5^m) is an integer?

So, in order for 10^m to divide into N, we need m factors of 2 and m factors of 5. Since we know there are fewer factors of 5 than 2 within the multiples of 3 between 1 and 100, let's determine the number of factors of 5 within the multiples of 3 between 1 and 100.

3 x 5 = 15 (1 factor of 5)

3 x 10 = 30 (1 factor of 5)

3 x 15 = 45 (1 factor of 5)

3 x 20 = 60 (1 factor of 5)

3 x 25 = 75 (2 factors of 5)

3 x 30 = 90 (1 factor of 5)

Thus, there are 7 factors of 5 within the multiples of 3 between 1 and 100, and thus the maximum value of m is 7.

Answer: C

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Founder and CEO
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