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If d is a positive integer and f is the product of the first

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If d is a positive integer and f is the product of the first [#permalink] New post 28 Jan 2012, 17:13
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If d is a positive integer and f is the product of the first 30 positive integers, what is the value of d?

(1) 10^d is a factor of f
(2) d>6
[Reveal] Spoiler: OA

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Last edited by Bunuel on 30 Oct 2012, 00:42, edited 2 times in total.
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Re: +ve integer D [#permalink] New post 28 Jan 2012, 17:18
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If d is a positive integer and f is the product of the first 30 positive integers, what is the value of d?

(1) 10^d is a factor of f --> k*10^d=30!.

First we should find out how many zeros 30! has, it's called trailing zeros. It can be determined by the power of 5 in the number 30! --> \frac{30}{5}+\frac{30}{25}=6+1=7 --> 30! has 7 zeros.

k*10^d=n*10^7, (where n is the product of other multiples of 30!) --> it tells us only that max possible value of d is 7. Not sufficient.

Side notes: 30! is some huge number with 7 trailing zeros (ending with 7 zeros). Statement (1) says that 10^d is factor of this number, but 10^d can be 10 (d=1) or 100 (d=2) ... or 10,000,000 (d=7). Basically d can be any integer from 1 to 7, inclusive (if d>7 then 10^d won't be a factor of 30! as 30! has only 7 zeros in the end). So we can not determine single numerical value of d from this statement. Hence this statement is not sufficient.

(2) d>6 Not Sufficient.

(1)+(2) From (2) d>6 and from (1) d_{max}=7 --> d=7.

Answer: C.

Hope it helps.
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Re: +ve integer D [#permalink] New post 28 Jan 2012, 17:19
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Trailing zeros:
Trailing zeros are a sequence of 0s in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

125000 has 3 trailing zeros;

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, can be determined with this formula:

\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}, where k must be chosen such that 5^(k+1)>n

It's more simple if you look at an example:

How many zeros are in the end (after which no other digits follow) of 32!?
\frac{32}{5}+\frac{32}{5^2}=6+1=7 (denominator must be less than 32, 5^2=25 is less)

So there are 7 zeros in the end of 32!

The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero.

For more on this concept check Everything about Factorials on the GMAT: everything-about-factorials-on-the-gmat-85592.html
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Re: If d is a positive integer and f is the product of the first [#permalink] New post 29 Jan 2012, 14:50
Hi Bunuel thanks - all makes sense apart from the concept of trailing zeros.

Am I right in saying this is how you said there will be 7 zero's.

30/5 + 30/25 = 6 + 1 (quotient) = 7. Where I am not clear is have you simply divided 30/25? I hope I am making myself clear, if I am not then please let me know.
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Re: If d is a positive integer and f is the product of the first [#permalink] New post 29 Jan 2012, 15:00
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enigma123 wrote:
Hi Bunuel thanks - all makes sense apart from the concept of trailing zeros.

Am I right in saying this is how you said there will be 7 zero's.

30/5 + 30/25 = 6 + 1 (quotient) = 7. Where I am not clear is have you simply divided 30/25? I hope I am making myself clear, if I am not then please let me know.


Yes, you take only the integer part. For example how many trailing zeros does 126! have?

126/5+126/5^2+126/5^3=25+5+1=31.

Hope it's clear.
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Re: If d is a positive integer and f is the product of the first [#permalink] New post 04 Jun 2013, 20:17
Hi. I think there is a solution without knowing the trailing zeros formula. Of course I. alone will not be enough (d= 10 or d=100 do the trick) and II. d>6 is vague. Now, to evaluate I and II together, like you know that 10 = 2*5, and 10^x = (2*5)^x = 2^x*5^x, if 10^d is a factor of f, like f = 1*2*3*4*5...*30, you need to see how many 2s and 5ves you can get. You have plenty of 2s, so lets focus on the 5ves. You actually get 7 fives between 1 and 30 (one in 5,10,15,20,30 and two in 25). So basically d could be any number between 1 and 7. Like II is "d>6", you know that d = 7.

(my 1st post, sorry for style... just trying to help, I suck at knowing formulas, although they save you time. (I learned a lot from this forum, Bunuel is my Guru!))
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Re: If d is a positive integer and f is the product of the first [#permalink] New post 04 Jun 2013, 21:25
I'm having trouble understanding "the product of the first 30 positive integers"
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Re: If d is a positive integer and f is the product of the first [#permalink] New post 04 Jun 2013, 23:19
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Yahtzeefish wrote:
I'm having trouble understanding "the product of the first 30 positive integers"


The product of the first 30 positive integers = 1*2*3*...*29*30=30!.

Check Factorials chapter of our Math Book for more: everything-about-factorials-on-the-gmat-85592.html

Hope it helps.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If d is a positive integer and f is the product of the first   [#permalink] 04 Jun 2013, 23:19
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