GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 06 Jul 2020, 20:08 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If x is the product of the integers from 1 to 150, inclusive

Author Message
TAGS:

### Hide Tags

Intern  Joined: 04 Jul 2013
Posts: 5
Location: United Kingdom
Concentration: Strategy, Finance
Schools: LBS '16 (A)
GMAT 1: 650 Q42 V38
GMAT 2: 690 Q45 V40
WE: Consulting (Investment Banking)
If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

22
91 00:00

Difficulty:   35% (medium)

Question Stats: 68% (01:06) correct 32% (01:55) wrong based on 1348 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: 65014
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

26
53
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).

_________________
Director  B
Joined: 03 Feb 2013
Posts: 833
Location: India
Concentration: Operations, Strategy
GMAT 1: 760 Q49 V44
GPA: 3.88
WE: Engineering (Computer Software)
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

5
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)
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 65014
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

12
22
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).

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.
_________________
Director  S
Joined: 08 Jun 2010
Posts: 661
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

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
Current Student B
Joined: 20 Jan 2017
Posts: 49
Location: United States (NY)
Schools: CBS '20 (A)
GMAT 1: 750 Q48 V44 GMAT 2: 610 Q34 V41
GPA: 3.92
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

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

Posted from my mobile device
Intern  B
Joined: 06 Mar 2012
Posts: 11
Schools: Booth '15
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 65014
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

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!.
_________________
Current Student V
Joined: 27 May 2014
Posts: 502
GMAT 1: 730 Q49 V41 Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

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.
_________________
Director  D
Joined: 13 Mar 2017
Posts: 713
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)
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

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

Senior SC Moderator V
Joined: 22 May 2016
Posts: 3940
If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

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:

_________________
Visit SC Butler, here! Get two SC questions to practice, whose links you can find by date.

Our lives begin to end the day we become silent about things that matter. -- Dr. Martin Luther King, Jr.

BLACK LIVES MATTER.
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 17031
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 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.

GMAT assassins aren't born, they're made,
Rich
_________________
SVP  V
Status: It's near - I can see.
Joined: 13 Apr 2013
Posts: 1676
Location: India
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 V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 11043
Location: United States (CA)
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

1
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!.

_________________

# 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

Intern  B
Joined: 27 Jul 2017
Posts: 45
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
Joined: 05 Feb 2018
Posts: 440
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: 17031
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
_________________
Non-Human User Joined: 09 Sep 2013
Posts: 15382
Re: If x is the product of the integers from 1 to 150, inclusive  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: If x is the product of the integers from 1 to 150, inclusive   [#permalink] 23 Jun 2020, 12:12

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