Last visit was: 15 Jul 2024, 13:49 It is currently 15 Jul 2024, 13:49
Close
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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
Math Expert
Joined: 02 Sep 2009
Posts: 94354
Own Kudos [?]: 641113 [53]
Given Kudos: 85011
Send PM
Most Helpful Reply
Director
Director
Joined: 01 Oct 2017
Status:Learning stage
Posts: 822
Own Kudos [?]: 1367 [14]
Given Kudos: 41
WE:Supply Chain Management (Energy and Utilities)
Send PM
MBA Section Director
Joined: 22 May 2017
Affiliations: GMATClub
Posts: 12761
Own Kudos [?]: 8913 [9]
Given Kudos: 3038
GRE 1: Q168 V154
GPA: 3.4
WE:Engineering (Education)
Send PM
General Discussion
Intern
Intern
Joined: 20 Feb 2017
Status:Rise above
Posts: 42
Own Kudos [?]: 84 [0]
Given Kudos: 37
Location: India
Schools: ESSEC '22
GMAT 1: 650 Q47 V34
GPA: 3
WE:Editorial and Writing (Entertainment and Sports)
Send PM
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
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
Director
Joined: 01 Oct 2017
Status:Learning stage
Posts: 822
Own Kudos [?]: 1367 [1]
Given Kudos: 41
WE:Supply Chain Management (Energy and Utilities)
Send PM
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
1
Bookmarks
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.
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11473
Own Kudos [?]: 34343 [3]
Given Kudos: 322
Send PM
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
1
Kudos
2
Bookmarks
Expert Reply
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
Intern
Joined: 15 Mar 2019
Posts: 17
Own Kudos [?]: 7 [0]
Given Kudos: 12
Send PM
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
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: 10135
Own Kudos [?]: 17018 [0]
Given Kudos: 4
GMAT 1: 760 Q51 V42
GPA: 3.82
Send PM
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
Expert Reply
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.
Manager
Manager
Joined: 10 Aug 2018
Posts: 227
Own Kudos [?]: 141 [0]
Given Kudos: 179
Location: India
Concentration: Strategy, Operations
WE:Operations (Energy and Utilities)
Send PM
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
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
VP
VP
Joined: 15 Dec 2016
Posts: 1352
Own Kudos [?]: 220 [0]
Given Kudos: 188
Send PM
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
VeritasKarishma - is there a blog post discussing this concept which is needed to solve this question ? (on the blog post : quarter wit blog post)
VP
VP
Joined: 15 Dec 2016
Posts: 1352
Own Kudos [?]: 220 [0]
Given Kudos: 188
Send PM
If x and y are both integers greater than 1, is xy>100 ? [#permalink]
Bunuel VeritasKarishma - i was wondering as I did this problem - if i could have this take-away

if the total number of factors is odd = perfect square
-- if the total number of factors is odd and number of factors is a prime number of factors (ex : 7 factors) = this perfect square has only one type of prime number in its prime box
-- if the total number of factors is odd and number of factors happens to be non-prime number of factors (Ex :9 factors) = this perfect square has \(\geq{1}\) type of prime number in its prime box

if the total number of factors is even = non perfect square
-- If the total number of factors is 2 = number has to be Prime. Number of types of prime numbers in the prime box = 1 type only (\(2^{1}\) or \(3^{1}\) or \(7^{1}\) or \(29^{1}\)- all have 2 factors only )
-- If the total number of factors is 4 = Number of types of prime numbers in the prime box \(\geq{1}\) type (\(2^{3}\) or \(2^{1}\) * \(3^{1}\), both have 4 factors each )
-- If the total number of factors is 8 = Number of types of prime numbers in the prime box \(\geq{1}\) type (\(2^{7}\) or \(2^{1}\) * \(3^{3}\) or \(2^{1}\) * \(3^{1}\) * \(5^{1}\), all three have 8 factors each )

Other take-aways

** If the total number of factors is 1 = number has to be 1
Tutor
Joined: 16 Oct 2010
Posts: 15110
Own Kudos [?]: 66650 [2]
Given Kudos: 436
Location: Pune, India
Send PM
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
2
Kudos
Expert Reply
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.


Check this first:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2010/1 ... ly-number/

(1) x has exactly 7 unique factors.

Total number of factors is given by (p+1)(q+1)...
If (p+1)(q+1)... = 7 = 1 * 7, this can only be obtained when p = 6.
The smallest value of x can be 2^6 = 64.
Now, since y is an integer greater than 1, it will be 2 or more. No matter what y is then, xy will certainly be more than 100.
Sufficient.

(2) y has exactly 9 unique factors.
If (p+1)(q+1)... = 9 = 1*8 or 3*3
The smallest value y can taken is when p and q are 2 each such that y = 2^2 * 3^2 = 36.
Then if x is 2, xy < 100.
If x = 3, xy > 100.
Not sufficient.

Answer (A)
User avatar
Non-Human User
Joined: 09 Sep 2013
Posts: 33980
Own Kudos [?]: 851 [0]
Given Kudos: 0
Send PM
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
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.
GMAT Club Bot
Re: If x and y are both integers greater than 1, is xy>100 ? [#permalink]
Moderator:
Math Expert
94354 posts