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

 It is currently 19 Oct 2019, 02:47 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 m and n are both two digit numbers and m-n = 11x, is x an

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

Hide Tags

Manager  Joined: 02 Jun 2011
Posts: 114
If m and n are both two digit numbers and m-n = 11x, is x an  [#permalink]

Show Tags

1
10 00:00

Difficulty:   65% (hard)

Question Stats: 61% (01:52) correct 39% (01:59) wrong based on 287 sessions

HideShow timer Statistics

If m and n are both two digit numbers and m-n = 11x, is x an integer?

(1) The tens digit and the units digit of m are same
(2) m+n is a multiple of 11
Math Expert V
Joined: 02 Sep 2009
Posts: 58452
Re: If m and n are both two digit numbers and m-n = 11x, is x an  [#permalink]

Show Tags

2
If m and n are both two digit numbers and m-n = 11x, is x an integer?

The question basically asks whether m-n is a multiple of 11.

(1) The tens digit and the units digit of m are same --> m could be: 11, 22, 33, ..., 99 --> m is a multiple of 11. Not sufficiient since no info about n.

(2) m+n is a multiple of 11 --> if m=n=11 then the m-n is a multiple of 11 but if m=12 and n=10 then m-n is NOT a multiple of 11. Not sufficient.

(1)+(2) From (1) we have that m={multiple of 11} and from (2) we have that m+n={multiple of 11} --> {multiple of 11}+n={multiple of 11} --> n={multiple of 11} --> m-n={multiple of 11}-{multiple of 11}={multiple of 11}. Sufficient.

Below might help to understand this concept better.

If integers $$a$$ and $$b$$ are both multiples of some integer $$k>1$$ (divisible by $$k$$), then their sum and difference will also be a multiple of $$k$$ (divisible by $$k$$):
Example: $$a=6$$ and $$b=9$$, both divisible by 3 ---> $$a+b=15$$ and $$a-b=-3$$, again both divisible by 3.

If out of integers $$a$$ and $$b$$ one is a multiple of some integer $$k>1$$ and another is not, then their sum and difference will NOT be a multiple of $$k$$ (divisible by $$k$$):
Example: $$a=6$$, divisible by 3 and $$b=5$$, not divisible by 3 ---> $$a+b=11$$ and $$a-b=1$$, neither is divisible by 3.

If integers $$a$$ and $$b$$ both are NOT multiples of some integer $$k>1$$ (divisible by $$k$$), then their sum and difference may or may not be a multiple of $$k$$ (divisible by $$k$$):
Example: $$a=5$$ and $$b=4$$, neither is divisible by 3 ---> $$a+b=9$$, is divisible by 3 and $$a-b=1$$, is not divisible by 3;
OR: $$a=6$$ and $$b=3$$, neither is divisible by 5 ---> $$a+b=9$$ and $$a-b=3$$, neither is divisible by 5;
OR: $$a=2$$ and $$b=2$$, neither is divisible by 4 ---> $$a+b=4$$ and $$a-b=0$$, both are divisible by 4.

Hope it's clear.
_________________
Intern  Joined: 07 Mar 2013
Posts: 24
Re: If m and n are both two digit numbers and m-n = 11x, is x an  [#permalink]

Show Tags

Hi Banuel,

one confusion here.

from (1) we know that m is a multiple of 11. we also know that m-n= multiple of 11.
now, if we consider m to be 99 than , 99-n=multiple of 11. can we have any other 2 digit no. which is NOT a multiple of 11 for n in this case ? I think no. so effectively shouldn't the answer be A ? Have I missed something here ?
Verbal Forum Moderator B
Joined: 10 Oct 2012
Posts: 590
Re: If m and n are both two digit numbers and m-n = 11x, is x an  [#permalink]

Show Tags

vishalrastogi wrote:
Hi Banuel,

one confusion here.

from (1) we know that m is a multiple of 11. we also know that m-n= multiple of 11.
now, if we consider m to be 99 than , 99-n=multiple of 11. can we have any other 2 digit no. which is NOT a multiple of 11 for n in this case ? I think no. so effectively shouldn't the answer be A ? Have I missed something here ?

The highlighted portion is the problem. When you say (m-n) is a multiple of 11, that implies that you ARE saying that x IS an integer. But that is something which is in-fact being asked. You can't assume it while solving for the question stem.
_________________
SVP  Joined: 06 Sep 2013
Posts: 1570
Concentration: Finance
Re: If m and n are both two digit numbers and m-n = 11x, is x an  [#permalink]

Show Tags

First we know that m is a a multiple of 11 but we still don’t know anything about 'n' therefore insufficient. Then we know that m+n is a multiple of 11 but that doesn't mean that they are both multiples of 11, it could be that they are both not non multiples of 11. Both together since m is a multiple of 11 then n must also be a multiple of 11. The difference of two multiples of 11 is always a multiple of 11. Thus answer is C

Hope this clarifies

Gimme some freaking Kudos!!
Best
J
Manager  Joined: 03 May 2013
Posts: 67
Re: If m and n are both two digit numbers and m-n = 11x, is x an  [#permalink]

Show Tags

how can we say that m is a multiple of 11

m could be 11.10
No where mentioned is the stem that M is an int , its just a number

OA should E
Math Expert V
Joined: 02 Sep 2009
Posts: 58452
Re: If m and n are both two digit numbers and m-n = 11x, is x an  [#permalink]

Show Tags

vipulgoel wrote:
how can we say that m is a multiple of 11

m could be 11.10
No where mentioned is the stem that M is an int , its just a number

OA should E

11.10 is not a 2-digit number.
_________________
GMAT Tutor G
Joined: 24 Jun 2008
Posts: 1806
Re: If m and n are both 2 digit numbers and m-n=11x, is x an integer?  [#permalink]

Show Tags

1
If a number is a multiple of 11, that means you can write the number as '11q' where q is an integer. So if m - n = 11x, and x is an integer, that just means that m-n is a multiple of 11. So that's the question we're trying to answer: is m - n divisible by 11?

Statement 1 tells us that the two-digit number m looks something like 11, 22, 33, etc. Those numbers are all multiples of 11, so Statement 1 is just telling us that m alone is divisible by 11. We know nothing about n, however, so we can't say if m - n is divisible by 11.

Statement 2 tells us that m + n is divisible by 11. This is not sufficient, as you can see by generating almost any two examples, one using multiples of 11, and one not. So maybe m = 22 and n = 11, and then m-n is also divisible by 11. But maybe m = 23 and n = 10, and then m - n is not divisible by 11.

Using both statements, we know that m+n and m are both multiples of 11. That guarantees that n is also a multiple of 11. And if m and n are both multiples of 11, then m-n will always be a multiple of 11 as well, so the answer is C.

If it's unclear why n must be a multiple of 11 here, you can see that in one of the following ways:

• perhaps the fastest is to use the fact that, if we subtract one multiple of 11 from another, we always get a multiple of 11. So if (m+n) and (m) are both multiples of 11, then (m+n) - m will be too, but that's just equal to n.

• the longer way is conceptually more useful to understand, because it illustrates why it's true that you always get a multiple of 11 when you add or subtract two multiples of 11:

- if m+n is a multiple of 11, then m+n = 11q for some integer q
- if m is a multiple of 11, then m = 11k for some integer k

So we know:

m + n = 11q

but m = 11k, so we can substitute for m and rearrange:

11k + n = 11q
n = 11q - 11k
n = 11 (q - k)

and we can now see that n is equal to 11 times some integer, so n is a multiple of 11 also.

Of course there's nothing special about '11' here. When you add or subtract two multiples of any integer p, you always get a multiple of p.
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
Non-Human User Joined: 09 Sep 2013
Posts: 13266
Re: If m and n are both two digit numbers and m-n = 11x, is x an  [#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 m and n are both two digit numbers and m-n = 11x, is x an   [#permalink] 13 Mar 2018, 07:00
Display posts from previous: Sort by

If m and n are both two digit numbers and m-n = 11x, is x an

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

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