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If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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21
71 00:00

Difficulty:   35% (medium)

Question Stats: 66% (01:09) correct 34% (01:56) wrong based on 1016 sessions

### HideShow timer Statistics If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39
Math Expert V
Joined: 02 Sep 2009
Posts: 56300
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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24
49
bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39

$$x=150!$$. We need to find the power of 5 in prime factorization of 150!.

150/5 + 150/5^2 + 150/5^3 = 30 + 6 + 1 = 37 (check here: everything-about-factorials-on-the-gmat-85592.html).

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Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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4
3
bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39

It basically asks for the number of 5s in 150!

150/5 + 150/25 + 150/125 = 30 + 6 + 1. Hence 37 Option d)
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##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 56300
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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10
17
Bunuel wrote:
bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39

$$x=150!$$. We need to find the power of 5 in prime factorization of 150!.

150/5 + 150/5^2 + 150/5^3 = 30 + 6 + 1 = 37 (check here: everything-about-factorials-on-the-gmat-85592.html).

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Hope it helps.
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Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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2
bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39

total number of 5 is 150/5=30
among 30 there are 25 50 75 100 125 150

contain 1,1,1,1 ,2 , 1 the number of 5 more

total 30+7

d
very hard
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Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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1) To paraphrase the question, we need to find all the prime factors 5 of the number 150!
2) 150/5=30; 150/25=6; 150/125=1. The total number of 5's is 30+6+1=37

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Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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Math Expert V
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Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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divyakesharwani wrote:
This is same as finding trailing zero - right.

Yes. The number of trailing zeros is equal to the number of power of 5 in n!.
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GMAT 1: 730 Q49 V41 Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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1
5^3 < 150 < 5^4

Hence, the total number of 5 in 150!:

150/5^1 + 150/5^2 + 150/5^3 = 30 + 6 + 1 = 37

So, y = 37
Ans: D.
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Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39

It is asking the number of 5 when the multiplication is written in terms of prime factors

So, y = [150/5] + [150/25] + [150/125] = 30 + 6 + 1 = 37

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If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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2
bgribble wrote:
If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ?

A) 30
B) 34
C) 36
D) 37
E) 39

Spelled out a little more:

1) $$x$$ = product of integers from 1 to 150
$$x$$ = 150 * 149* 148 . . .* 3 * 2 *1:
That is, $$x$$ = 150!

2) $$5^{y}$$ is a factor of 150! What is the greatest possible value of $$y$$?

Using $$\frac{150}{5^{y}}$$, consider each power $$y$$, of 5. Do not worry about remainders.

$$\frac{150}{5^1}$$ = 30
(5 divides into 150 thirty times)

$$\frac{150}{5^2}$$ = 6
(25 divides into 150 six times)

$$\frac{150}{5^3}$$ = 1
(125 divides into 150 only once. Ignore the remainder.)

$$5^4 = 625$$ -- too large to divide into 150 as a factor.

3) Add the results: 30 + 6 + 1 = 37

Once you know the theory and method, questions such as this one are pretty straightforward. The stats here might indicate that the suggestion below is indispensable.

Bunuel wrote:
Quote:

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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

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Hi All,

Since we're multiplying a big string of numbers together, this question comes down to "prime factorization"....we need to "find" all of the 5s that exist in this string of numbers. As a hint, some numbers have MORE THAN one 5 in them.

To start, we know that there are 30 multiples of 5 in the string from 1 to 150, so that's 30 5s right there.

Now, we need to think about numbers that have more than one 5 in them....

5, 10, 15....these all have just one 5

25, 50, 75, 100, 150...these all have TWO 5s; we already counted one of the 5s in each, so we have to now add the other one to the total = +5 more

125....this has THREE 5s; we already counted one of the 5s, so we have to now add the other two to the total = +2 more

30 + 5 + 2 = 37 fives.

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ SVP  V Status: It's near - I can see. Joined: 13 Apr 2013 Posts: 1686 Location: India Concentration: International Business, Operations Schools: INSEAD Jan '19 GPA: 3.01 WE: Engineering (Real Estate) If x is the product of the integers from 1 to 150, inclusive [#permalink] ### Show Tags bgribble wrote: If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ? A) 30 B) 34 C) 36 D) 37 E) 39 x is the product of the integers from 1 to 150, inclusive means x = 150! 5^y is a factor of 150! means, $$\frac{150!}{5^y}$$ $$= I$$, where "I" is an integer We need to calculate the no. of 5s in 150! = $$\frac{150}{5} + \frac{150}{25} + \frac{150}{125}$$ = $$30 + 6 + 1$$ = $$37$$ (D) _________________ "Do not watch clock; Do what it does. KEEP GOING." Target Test Prep Representative D Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 6937 Location: United States (CA) Re: If x is the product of the integers from 1 to 150, inclusive [#permalink] ### Show Tags bgribble wrote: If x is the product of the integers from 1 to 150, inclusive, and 5^y is a factor of x, what is the greatest possible value of y ? A) 30 B) 34 C) 36 D) 37 E) 39 The product of the integers from 1 to 150, inclusive, is 150!. To determine the number of factors of 5 within 150!, we can use the following shortcut in which we divide 150 by 5, and then divide the quotient of 150/5 by 5 and continue this process until we can no longer get a nonzero integer as the quotient. 150/5 = 30 30/5 = 6 6/5 = 1 (we can ignore the remainder) Since 1/5 does not produce a nonzero quotient, we can stop. The final step is to add up our quotients; that sum represents the number of factors of 5 within 150!. Thus, there are 30 + 6 + 1 = 37 factors of 5 within 150!. Answer: D _________________ # Scott Woodbury-Stewart Founder and CEO Scott@TargetTestPrep.com See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Manager  B Joined: 27 Jul 2017 Posts: 50 Re: If x is the product of the integers from 1 to 150, inclusive [#permalink] ### Show Tags The answer must be D, i.e. 5 will have a total power of 37 in 150!. _________________ Ujjwal Sharing is Gaining! Senior Manager  P Status: Gathering chakra Joined: 05 Feb 2018 Posts: 367 If x is the product of the integers from 1 to 150, inclusive [#permalink] ### Show Tags Bunuel EMPOWERgmatRichC generis When is it possible to get extra factors of 5 (or other numbers)? I recall doing a similar problem where there is an exception to this rule of just dividing by increasing amounts. I can't find it, but I think there was some extra factors because it involved adding 2 numbers (probably factorials), which added up to have 2 additional factors of 5 or something. EMPOWERgmat Instructor V Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 14590 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If x is the product of the integers from 1 to 150, inclusive [#permalink] ### Show Tags 1 Hi energetics, In simple terms, you would look for squares, cubes, quads, etc. that divide evenly into the larger product. For example, in this prompt: 25 = (25)(1) = 5^2 50 = (25)(2) = (5^2)(2) 75 = (25)(3) = (5^2)(3) ... 125 = (25)(5) = (5^2)(5) = 5^3 Etc. IF a question asks you to combine two numbers (through addition or multiplication), then there will almost always be some additional factors to account for (otherwise, why would the question ask you to do the additional 'math work'?). Remember that NOTHING about a GMAT question is ever 'random' - each question was written by a human to 'test' you on specific concepts, so one of the most important questions you can ever ask yourself when you're working through a Quant or Verbal question is "why was I given this information (because I was given it for some specific reason)?" GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** # Rich Cohen Co-Founder & GMAT Assassin Follow Special Offer: Save$75 + GMAT Club Tests Free
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