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

 It is currently 12 Nov 2019, 14:33

### 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

# If x and y are both integers greater than 1, is xy>100 ?

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

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 58991
If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

### Show Tags

26 Jul 2018, 21:12
18
00:00

Difficulty:

95% (hard)

Question Stats:

26% (02:20) correct 74% (01:49) wrong based on 163 sessions

### HideShow timer Statistics

If x and y are both integers greater than 1, is xy > 100?

(1) x has exactly 7 unique factors.

(2) y has exactly 9 unique factors.

_________________
##### Most Helpful Community Reply
Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 995
WE: Supply Chain Management (Energy and Utilities)
If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

### Show Tags

26 Jul 2018, 23:20
4
3
Bunuel wrote:
If x and y are both integers greater than 1, is xy > 100?

(1) x has exactly 7 unique factors.

(2) y has exactly 9 unique factors.

Given situation:- $$x>1, y>1$$ (x and y are integers)
Stem:- Is $$xy>100$$

St1:- x has exactly 7 unique factors
Or, we can write in the prime factorization form,$$x=a^p$$, where p+1=7 or, p=6 (I have taken x=a^p not x=a^p*b^q..., because we are given 7 factors ,if we consider more than one prime exponent, we can't get a multiplication result of 7 by multiplying more than one integer values(where none of the integers are 1)) or $${(p+1)(q+1)...}\neq7$$
So,$$x=a^6$$
Now $$minimum(x)=min(a^6)=2^6=64$$ (we can't consider a=1 since it yields x=1^6=1)
Therefore, $$x*y=64*2=128>100$$ (The lowest integer value of y greater than 1 is 2)
For all other positive integer values of a>2,we have x > 100. Subsequently $$xy>100$$.

Sufficient.

St2:-y has exactly 9 unique factors
With the same reasoning as stated in st1, we have $$y=a^p*b^q$$, where (p+1)(q+1)=9=3*3.
So, p+1=3 and q+1=3
Or, p=2 and q=2
So, $$min(y)=min(a^2*b^2)=2^2*3^2=36$$
Therefore, $$xy=2*36=72<100$$----------------(a) (when x=2)
and $$xy=3*36=108>100$$-----------------------(b) (when x=3)
From (a) and (b), st2 is insufficient since st2 is inconsistent with the question stem.

Ans. (A)
_________________
Regards,

PKN

Rise above the storm, you will find the sunshine
##### General Discussion
MBA Section Director
Affiliations: GMATClub
Joined: 22 May 2017
Posts: 2648
GPA: 4
WE: Engineering (Computer Software)
If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

### Show Tags

26 Jul 2018, 21:34
2
Any integer with odd number of unique factors is a perfect square

If x and y are both integers greater than 1

To determine whether xy > 100 ?

Statement 1

x has exactly 7 unique factors.

=> x is a perfect square

=> x $$\geq$$ 64 since no perfect squate less than 64 has 7 number of factors

=> Since y is an integer and y > 1 => xy $$\geq$$ 128

Statement 1 is sufficient

Statement 2

y has exactly 9 unique factors.

=> y is a perfect square

=> y $$\geq$$ 36 since no perfect square less than 36 has 9 factors

=> Since x is an integer and x > 1 => xy $$\geq$$ 72

Statement 2 is not sufficient

Hence option A
_________________
Intern
Status: Rise above
Joined: 20 Feb 2017
Posts: 45
Location: India
Schools: ESSEC '22
GMAT 1: 650 Q47 V34
GPA: 3
WE: Editorial and Writing (Entertainment and Sports)
Re: If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

### Show Tags

29 Jul 2018, 06:51
Correct me if I'm wrong but doesn't 24 also have 7 factors, making statement 1 insufficient? I am a bit unsure about this. Hence I went ahead with option C.
Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 995
WE: Supply Chain Management (Energy and Utilities)
Re: If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

### Show Tags

29 Jul 2018, 07:05
1
nithinjohn wrote:
Correct me if I'm wrong but doesn't 24 also have 7 factors, making statement 1 insufficient? I am a bit unsure about this. Hence I went ahead with option C.

Hi nithinjohn ,

$$24=2^3*3^1$$
So the no of factors=(3+1)*(1+1)=8

And the factors are 1,2,3,4,6,8,12, and 24.

https://gmatclub.com/forum/divisibility ... 74998.html
You may visit above link for more clarity.

FINDING THE NUMBER OF FACTORS OF AN INTEGER
First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

This is available in the link provided.
_________________
Regards,

PKN

Rise above the storm, you will find the sunshine
Math Expert
Joined: 02 Aug 2009
Posts: 8150
Re: If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

### Show Tags

29 Jul 2018, 08:02
1
2
Bunuel wrote:
If x and y are both integers greater than 1, is xy > 100?

(1) x has exactly 7 unique factors.

(2) y has exactly 9 unique factors.

xy>100 and both x and y>1. MEANS both are at least 2 and therefore if we can prove at least one is >50, it would be sufficient

1) x has exactly 7 unique factors
a) 7=1*7, no other possibilities so it is a^6
b) odd number of factor means X is perfect square
c) if it were just 3 factors.. it meant a perfect square of prime number
So least value =2^6=64
So minimum value of xy is 2*64=128>50
Sufficient

2) y has 9 unique factors.
a) higher number of factors necessarily does not mean LARGER number
b) 9=1*9....so least value 2^8=256>50 ....yes
c) 9=3*3.... So type a^2*b^2 least value = 2^2*3^2=4*9=36>50....No
So xy>100 and xy<100 possible
Insufficient

A
_________________
Intern
Joined: 15 Mar 2019
Posts: 19
Re: If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

### Show Tags

16 Aug 2019, 11:37
Hi guys,

im just starting out on this topic, but i am very confused.
The Tasks states "unique" factors, so finding the first 7 UNIQUE factors and multiply them results always in >100 when y>=1 ?
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8125
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

### Show Tags

16 Aug 2019, 14:06
Bunuel wrote:
If x and y are both integers greater than 1, is xy > 100?

(1) x has exactly 7 unique factors.

(2) y has exactly 9 unique factors.

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables ($$x$$ and $$y$$) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Conditions 1) & 2)
Since the integer x is $$p^6$$ type of integer greater than 1 by condition 1), where $$p$$ is a prime integer, the smallest possible number of is $$2^6 = 64$$.
Since the integer y is $$p^8$$ or $$p^2*q^2$$ types of integers greater than 1 condition 2), where $$p$$ and $$q$$ are prime numbers, the smallest possible number of y is $$2^2*3^2 = 36$$.
Then, the smallest possible value of $$xy$$ is $$64*36 > 100$$ and the answer is 'yes'.
Thus, both conditions together are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since the integer x is $$p^6$$ type of integer greater than 1 by condition 1), where $$p$$ is a prime integer, the smallest possible number of is $$2^6 = 64$$.
Since the smallest possible number of $$y$$ is 2, the smallest possible number of $$xy$$ is 128, which is greater than 100 and the answer is yes.
That's why condition 1) is sufficient.

Condition 2)
Since the integer y is $$p^8$$ or $$p^2*q^2$$ types of integers greater than 1 condition 2), where $$p$$ and $$q$$ are prime numbers, the smallest possible number of y is $$2^2*3^2 = 36$$.
If $$x = 2$$ and $$y = 36$$, then $$xy = 72$$ is less than 100 and the answer is "no".
If $$x = 3$$ and $$y = 36$$, then $$xy = 108$$ is greater than 100 and the answer is "yes".
Since condition 2) doesn't yield a unique answer, it is not sufficient.

Therefore, A is the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only \$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Senior Manager
Joined: 10 Aug 2018
Posts: 338
Location: India
Concentration: Strategy, Operations
WE: Operations (Energy and Utilities)
Re: If x and y are both integers greater than 1, is xy>100 ?  [#permalink]

### Show Tags

17 Aug 2019, 04:17
Oh now I get it.
Nice explanation.
Thanks.

chetan2u wrote:
Bunuel wrote:
If x and y are both integers greater than 1, is xy > 100?

(1) x has exactly 7 unique factors.

(2) y has exactly 9 unique factors.

xy>100 and both x and y>1. MEANS both are at least 2 and therefore if we can prove at least one is >50, it would be sufficient

1) x has exactly 7 unique factors
a) 7=1*7, no other possibilities so it is a^6
b) odd number of factor means X is perfect square
c) if it were just 3 factors.. it meant a perfect square of prime number
So least value =2^6=64
So minimum value of xy is 2*64=128>50
Sufficient

2) y has 9 unique factors.
a) higher number of factors necessarily does not mean LARGER number
b) 9=1*9....so least value 2^8=256>50 ....yes
c) 9=3*3.... So type a^2*b^2 least value = 2^2*3^2=4*9=36>50....No
So xy>100 and xy<100 possible
Insufficient

A

Posted from my mobile device
_________________
On the way to get into the B-school and I will not leave it until I win. WHATEVER IT TAKES.

" I CAN AND I WILL"

GMAT:[640 Q44, V34, IR4, AWA5]
Re: If x and y are both integers greater than 1, is xy>100 ?   [#permalink] 17 Aug 2019, 04:17
Display posts from previous: Sort by

# If x and y are both integers greater than 1, is xy>100 ?

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

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