Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 19 Jul 2019, 16:04

### 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
Your Progress

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

# What is the smallest positive integer x such that 450x is the cube of

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 56300
What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

19 Nov 2014, 08:08
00:00

Difficulty:

15% (low)

Question Stats:

73% (01:03) correct 27% (01:04) wrong based on 258 sessions

### HideShow timer Statistics

Tough and Tricky questions: Number Properties.

What is the smallest positive integer x such that 450x is the cube of a positive integer?

A. 2
B. 15
C. 30
D. 60
E. 120

Kudos for a correct solution.

_________________
Manager
Joined: 21 Jul 2014
Posts: 122
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

19 Nov 2014, 10:04
4
Hi Bunuel,

Great question. The way I'd suggest to solve this problem is to just Dive In.

We want to know the smallest x that will make 450x a CUBE of some number. Let's call that number y.

Let's first figure out what we're working with. The prime factorization of 450 can be visualized:
...........450
......../.......\
......45.......10
...../..\....../...\
...15...3...2.....5
.../..\
..5....3

So, we have 5 * 5 * 3 * 3 * 2 that can be multiplied together to get 450. Now we need to figure out what we need to make 450 * x into a cube of y (y^3=450*x).

We have two 5s, two 3s, and one 2. To arrange these numbers in identical triples (2,3,5), we need at least one more 5, one 3, and two 2's. Each of these triples will give us the value of y (2*3*5=30), which, multiplied by itself three times, gives us 450 * x.

Looking at the factors we need to complete the triples, we get 5 * 3 * 2 * 2 = 60. We know this is the smallest number possible because prime factors by definition cannot be broken down any further.

Therefore, we can go with answer choice D.

If time permits, we can do a sanity check. We calculated that y should be 2 * 3 * 5, or 30. 30 * 30 * 30 = 27000. Also, 450 * 60 = 27000.

Hope this helps!
Intern
Joined: 25 May 2014
Posts: 46
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

19 Nov 2014, 13:09
LighthousePrep wrote:
Hi Bunuel,

Great question. The way I'd suggest to solve this problem is to just Dive In.

We want to know the smallest x that will make 450x a CUBE of some number. Let's call that number y.

Let's first figure out what we're working with. The prime factorization of 450 can be visualized:
...........450
......../.......\
......45.......10
...../..\....../...\
...15...3...2.....5
.../..\
..5....3

So, we have 5 * 5 * 3 * 3 * 2 that can be multiplied together to get 450. Now we need to figure out what we need to make 450 * x into a cube of y (y^3=450*x).

We have two 5s, two 3s, and one 2. To arrange these numbers in identical triples (2,3,5), we need at least one more 5, one 3, and two 2's. Each of these triples will give us the value of y (2*3*5=30), which, multiplied by itself three times, gives us 450 * x.

Looking at the factors we need to complete the triples, we get 5 * 3 * 2 * 2 = 60. We know this is the smallest number possible because prime factors by definition cannot be broken down any further.

Therefore, we can go with answer choice D.

If time permits, we can do a sanity check. We calculated that y should be 2 * 3 * 5, or 30. 30 * 30 * 30 = 27000. Also, 450 * 60 = 27000.

Hope this helps!

Hi LighthousePrep,

I did the same way till y = 2*3*5=30 in order to complete the triplet as $$3^3 * 5^3 * 2^3$$.
but i did not get the below part:
Looking at the factors we need to complete the triples, we get 5 * 3 * 2 * 2 = 60. We know this is the smallest number possible because prime factors by definition cannot be broken down any further.
please help.
_________________
Never Try Quitting, Never Quit Trying
Intern
Joined: 10 Mar 2014
Posts: 36
Concentration: General Management, Technology
GMAT 1: 650 Q47 V32
GPA: 4
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

19 Nov 2014, 20:35
1
450 = 2 x 3^2 x 5^2 now we need two 2s, one 3 and one 5 to make it perfect cube.
So x= 2^2 x 3 x 5 = 60.

Answer is C.
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1787
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

19 Nov 2014, 20:48
2
Answer = C = 60

$$450 = 2^1 * 3^2 * 5^2$$

To make it a cube, all should have power of 3

$$x = 2^2 * 3^1 * 5^1 = 60$$
_________________
Kindly press "+1 Kudos" to appreciate
Intern
Joined: 20 Jan 2013
Posts: 33
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

19 Nov 2014, 21:13
1
Let's factorize 450

450 = $$2^{1}$$ * $$3^{2}$$ * $$5^{2}$$

Cube has all the powers of prime integers in multiple of 3, so to make 450 a cube we need to multiply it with the below number

x = $$2^{2}$$ * 3 * 5 = 60

Answer is D
_________________
_________________

KUDOS is always a good way to thank anyone.
It encourages someone to post more questions and also answers.

KUDOS Please if My post helps you in anyway. Your Kudos keep me alive [/color]
Manager
Joined: 21 Jan 2014
Posts: 61
WE: General Management (Non-Profit and Government)
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

20 Nov 2014, 02:44
since 450x is a cube of a positive no.
so 450x=2*5*5*3*3=2^1*5^2*3^2*x

hence 450x to be a cube ,x must b=2^2*5*3=60

Answer is D
Intern
Status: Struggling with verbal. Specially RC :-(
Joined: 10 Oct 2014
Posts: 2
Location: India
Concentration: Marketing, Strategy
WE: Sales (Manufacturing)
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

20 Nov 2014, 04:20
2
Above mentioned solutions are good. Let me try a GMAT focused solution to above problem.

As per the question:
Cube root of (450 multiplied by some number x) yields a positive integer.
i.e. (450 * x)^1/3

Now since, last digit of (multiplying 450 with any integer) will be zero, the options A and B are ruled out. Now we are left with options C, D, E.
It is always good to do such sanity checks as it helps eliminate wrong answers and thereby increasing the probability of right answer.

Now checking for each solution:
C. 450 * 30 or we can try 45*3 = 135 [Not even near to a cube root of any integer].
D. 450 * 60 or we can try 45*6 = 270 [Here we can see 27 is a cube root of 3. So 450*60=27000 which is a cube root of 30]
E. 450 * 120 or we can try 45*12 = 540 [Not even near to a cube root of any integer].

So correct answer is D.

Remember: Its always good to do sanity checks which will always help to get you to the right answer in GMAT.

This is my first explanation to any question. If you like this, press Kudos and do encourage.

Wishing good luck to all.
Math Expert
Joined: 02 Sep 2009
Posts: 56300
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

20 Nov 2014, 09:00
Official Solution:

What is the smallest positive integer $$x$$ such that $$450x$$ is the cube of a positive integer?

A. 2
B. 15
C. 30
D. 60
E. 120

The brute-force approach would be to systematically list multiples of 450 from 450 on up, test each one to see whether it is a perfect cube (the cube of a positive integer), and choose the first multiple that meets the criterion. However, this approach is very cumbersome. Even just trying the answer choices would take a long time. In fact, without insight into the nature of cubes, it is difficult to see how we can easily test whether a number is a cube, except by cubing various integers and comparing the results to the number in question.

A more efficient approach takes advantage of a key property of perfect cubes: its prime factors come in triplets. In other words, each of its prime factors occurs 3 times (or 6 times, 9 times, etc.) in the cube's prime factorization. To see why, try cubing $$6 = (2\times3)$$:
$$6 \times 6 \times 6 = (2 \times 3)(2 \times 3)(2 \times 3) = (2 \times 2 \times 2)(3 \times 3 \times 3)$$

As you can see, the 2's and 3's occur in triplets. So our goal is to make the prime factors of $$450x$$ occur in triplets as well.

The first step is to break up 450 into its prime factors:
$$450 = (45)(10) = (3 \times 3 \times 5)(2 \times 5) = 2 \times 3 \times 3 \times 5 \times 5$$

How many of each prime factor do we need to complete all the triplets? We are evidently missing two 2's, one 3, and one 5. Multiplying these missing factors together, we get $$2 \times 2 \times 3 \times 5 = 60$$.

Answer: D.
_________________
Manager
Joined: 11 Sep 2013
Posts: 139
Concentration: Finance, Finance
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

27 Nov 2014, 12:24
It is very easy question.

We need cube of 450x
Let a^3=450x
Prime factors of x = 5^2*3^2* 2
To make the number cube we need three 5, three 3 and three 2
So we need 5*3*2^2 = 60
Manager
Joined: 12 Sep 2014
Posts: 142
Concentration: Strategy, Leadership
GMAT 1: 740 Q49 V41
GPA: 3.94
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

28 Nov 2014, 14:55
For this problem, determine the prime factorization of 450. This yields two factors of both 3 and 5 with one factor of 2. To get the cube of an integer, simply add two factors of 2 and a factor each of 3 and 5. Multiplying this out yield 60. Answer is D.
Board of Directors
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4512
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

17 Jan 2017, 08:12
Bunuel wrote:

Tough and Tricky questions: Number Properties.

What is the smallest positive integer x such that 450x is the cube of a positive integer?

A. 2
B. 15
C. 30
D. 60
E. 120

Kudos for a correct solution.

$$450 = 2^1 * 3^2 * 5^2$$

So, we are $$2^2*3*5 = 60$$ short of a perfect cube...

Hence, in order that 450x is a perfect cube , x must be 60

Thus, Answer will be (D) 60
_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )
Manager
Joined: 25 Mar 2013
Posts: 235
Location: United States
Concentration: Entrepreneurship, Marketing
GPA: 3.5
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

19 Jan 2017, 15:44
450x = (y)^3
split 450
5x9x10xX = y^3
x = $$\frac{y^3}{5^2*3^2*2}$$
Check prime numbers each option 450x
We miss 60 for perfect cube
D
_________________
I welcome analysis on my posts and kudo +1 if helpful. It helps me to improve my craft.Thank you
Director
Joined: 02 Sep 2016
Posts: 657
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

03 Apr 2017, 11:17
Cube of an integer that means the power of each prime number must be 3 or a multiple of 3.

450=45*10=2*3^2*5^2

Thus we need 2^2*3^1*5^1=60
_________________
Help me make my explanation better by providing a logical feedback.

If you liked the post, HIT KUDOS !!

Don't quit.............Do it.
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 6937
Location: United States (CA)
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

06 Apr 2017, 09:53
Bunuel wrote:

Tough and Tricky questions: Number Properties.

What is the smallest positive integer x such that 450x is the cube of a positive integer?

A. 2
B. 15
C. 30
D. 60
E. 120

We must remember that all perfect cubes break down to unique prime factors, each of which has an exponent that is a multiple of 3. So, let’s break down 450 into primes to help determine what extra prime factors we need to make 450x a perfect cube.

450 = 45 x 10 = 9 x 5 x 5 x 2 = 3 x 3 x 5 x 5 x 2 = 5^2 x 3^2 x 2^1

In order to make 450x a perfect cube, we need two more 2s, one more 3, and one more 5. Thus, the smallest perfect cube that is a multiple of 450 is 5^3 x 3^3 x 2^3. In other words, 450x = (5^3)(3^3)(2^3). Thus:

x = (5^3 * 3^3 * 2^3)/450

x = (5^3 * 3^3 * 2^3)/(5^2 * 3^2 * 2^1)

x = 5^1 * 3^1* 2^2 = 60

Answer: D
_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
122 Reviews

5-star rated online GMAT quant
self study course

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.

Non-Human User
Joined: 09 Sep 2013
Posts: 11710
Re: What is the smallest positive integer x such that 450x is the cube of  [#permalink]

### Show Tags

25 Sep 2018, 04:31
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: What is the smallest positive integer x such that 450x is the cube of   [#permalink] 25 Sep 2018, 04:31
Display posts from previous: Sort by

# What is the smallest positive integer x such that 450x is the cube of

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne