GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 19 Oct 2019, 12:51

### 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

# x is the product of each integer from 1 to 50, inclusive and y=100^k,

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 58445
x is the product of each integer from 1 to 50, inclusive and y=100^k,  [#permalink]

### Show Tags

14 Sep 2015, 05:11
2
13
00:00

Difficulty:

65% (hard)

Question Stats:

54% (01:37) correct 46% (01:49) wrong based on 254 sessions

### HideShow timer Statistics

x is the product of each integer from 1 to 50, inclusive and y=100^k, where k is an integer . What is the greatest value of k for which y is a factor of x?

(A) 0
(B) 5
(C) 6
(D) 10
(E) 12

Kudos for a correct solution.

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 58445
Re: x is the product of each integer from 1 to 50, inclusive and y=100^k,  [#permalink]

### Show Tags

20 Sep 2015, 09:14
Bunuel wrote:
x is the product of each integer from 1 to 50, inclusive and y=100^k, where k is an integer . What is the greatest value of k for which y is a factor of x?

(A) 0
(B) 5
(C) 6
(D) 10
(E) 12

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

Before we even attack this problem, let’s discuss a drill to help you master the art of prime factorization and maximize your recognition of prime factors – and we’ll then show you how that drill will greatly improve your ability to attack problems like the above.

Prime Counting Drill:

Count from 1 to 50, hitting each integer in that range, but only allow yourself to use prime numbers. For example, the first 10 numbers are not:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

1, 2, 3, (2^2), 5, (2 x 3), 7, (2^3), (3^2), (2 x 5)

By doing this drill, you’ll force yourself to think in terms of prime factors and learn to quickly deconstruct a number into its prime, essential components. This ability – and this recognition – is hugely helpful on problems like the above. There, we’re asked to find out how large y can be and still be a factor of x. And since y is a placeholder value for 100^k, ultimately we’re being asked how many times 50! can be evenly divided by 100.

When approaching complex factor questions, prime factors are the name of the game. You should recognize that 100 is composed of the prime factors 2 x 2 x 5 x 5, so each time we can pair two 5s with two 2s we can divide by 100. And having performed the drill above you should see how we’ll obtain those 2s and 5s:

1, 2, 3, (2^2), 5, (2 x 3), 7, (2^3), )3^2), (2 x 5)

As you’ll see here, there are 2s aplenty – within the first 10 terms of 50! we have eight factors of 2 and only two factors of 5. So 5s will be our limiter, and we’ll look for 5s the same way that we’ve been thinking about numbers throughout this post. We’ll break out any multiple of 5 into its prime factors so that we can pull out all the 5s to maximize the value of 100^k.

5

10 = 2 x 5

15 = 3 x 5

20 = 2 x 2 x 5

25 = 5 x 5

30 = 2 x 3 x 5

35 = 7 x 5

40 = 2 x 2 x 2 x 5

45 = 3 x 3 x 5

50 = 2 x 5 x 5

There are 12 factors of 5 contained within 50!, and since we need 2 for each factor of 100 (and, as we’ve said, there are plenty of 2s to go around), there are 6 sets of 2 x 2 x 5 x 5, and the correct answer here is 6. And you can see, by training yourself to think in terms of prime factors, you have the full skill set available to you – plus the presence of mind to go with it – to attack a difficult question like this.

Recognizing and processing prime factors is a critical skill for high-level success on the GMAT, so you should train yourself to think through this lens. You may recall when you learned to type – for many of us, our initial typing class and its corresponding drills meant that nearly every time you heard a word, you’d mentally flip through a keyboard thinking of how to type it. For prime factorization you should think the same way; when you wake up on Sunday, July 24, you should almost immediately think: 7/(2 x 2 x 2 x 3). Sure, it’s annoying, but like it happened with typing this too will pass and the productive mindset will last. Hopefully it leads to something greater than 2 x 2 x 5 x 5 x 7…
_________________
##### General Discussion
CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2978
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Re: x is the product of each integer from 1 to 50, inclusive and y=100^k,  [#permalink]

### Show Tags

14 Sep 2015, 06:06
Bunuel wrote:
x is the product of each integer from 1 to 50, inclusive and y=100^k, where k is an integer . What is the greatest value of k for which y is a factor of x?

(A) 0
(B) 5
(C) 6
(D) 10
(E) 12

Kudos for a correct solution.

x = 1*2*3*4*5*......*49*50 = 50!
y = 100^k

100 = 10^2 = 2^2 * 5^2

On prime factorization of x, power of 5 will always be greater than power of 2

CONCEPT: In every Factorial value (x!), On prime factorization, the power of Bigger Prime number > the power of Smaller Prime number

Also, Power of any Prime Number in any factorial can be calculated by following understanding

Power of prime x in n! = [n/x] + [n/x^2] + [n/x^3] + [n/x^4] + ... and so on
Where,
[x] = Greatest Integer less than or equal to x and the explanation of terms is as follows
[n/x] = No. of Integers that are multiple of x from 1 to n
[n/x^2] = No. of Integers that are multiple of x^2 from 1 to n whose first power has been counted in previous step and second is being counted at this step
[n/x^3] = No. of Integers that are multiple of x^3 from 1 to n whose first two powers have been counted in previous two step and third power is counted at this step
And so on.....

Where [n/x] is greatest Integer value of (n/x) less than or equal to (n/x)
i.e. [100/3] = [33.33] = 33
i.e. [100/9] = [11.11] = 11 etc.

Power of 5 in x = [50/5] + [50/5^2] + [50/5^3]... = 10 + 2 + 0 = 12
Power of 2 in x = [50/2] + [50/2^2] + [50/2^3]... = 25 + 12 + 6 + 3 + 1 = 47

i.e. x = 50! = 2^47 * 5^12 * ... = 10^12 * ... = 100^6 * ...

i.e. x is divisible by a 6th power of 100 at the most
i.e. y = 100^k = 100^6
i.e. k = 6

_________________
Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION
Intern
Joined: 03 Feb 2014
Posts: 38
Location: United States
Concentration: Entrepreneurship, General Management
WE: General Management (Other)
Re: x is the product of each integer from 1 to 50, inclusive and y=100^k,  [#permalink]

### Show Tags

14 Sep 2015, 11:27
1
1
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:
n5+n52+n53+...+n5k, where k must be chosen such that 5k≤n
x = 1*2*3....*50 = 50!
No. of trailing zeros in 50! = 50/5 + 50/5^2 = 10+2 = 12
100^k = 10^2k → k = 12/2 = 6
_________________
--Shailendra
Senior Manager
Status: Countdown Begins...
Joined: 03 Jul 2016
Posts: 277
Location: India
Concentration: Technology, Strategy
Schools: IIMB
GMAT 1: 580 Q48 V22
GPA: 3.7
WE: Information Technology (Consulting)
x is the product of each integer from 1 to 50, inclusive and y=100^k,  [#permalink]

### Show Tags

08 Jan 2017, 01:53
The question is basically asking us to find how many 100's can be formed with product 1 to 50.

100 = 5^2*2^2

We have many 2^2 in the product of 1 to 50. So we have to find how many 5^2 we have in the product.
Total number of 5's we have are 10+2 =12. So 5^2 available are - 6.

Hope it helps..

Bunuel , please correct my understanding if it is wrong.
Non-Human User
Joined: 09 Sep 2013
Posts: 13275
Re: x is the product of each integer from 1 to 50, inclusive and y=100^k,  [#permalink]

### Show Tags

16 Jan 2018, 08:09
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: x is the product of each integer from 1 to 50, inclusive and y=100^k,   [#permalink] 16 Jan 2018, 08:09
Display posts from previous: Sort by