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

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Manager
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If m and n are both two digit numbers and m-n = 11x, is x an [#permalink]  30 May 2012, 12:36
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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
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 27000
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Kudos [?]: 40036 [2] , given: 5397

Re: If m and n are both two digit numbers and m-n = 11x, is x an [#permalink]  31 May 2012, 01:24
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Expert's post
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.
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Manager
Joined: 01 Sep 2012
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Kudos [?]: 38 [0], given: 19

If M and N are both two digit numbers and M-N=11X, is X an [#permalink]  14 Nov 2012, 07:27
If M and N are both two digit numbers and M-N=11X, is X an integer?
(1) The tens digit and units digit of M are the same
(2) M+N is a multiple of 11

I have a problem with this.
Two digit numbers means 13, 24... OR can it also be 22.1 (with decimal/fraction)?

I dont have the OA and thats why im asking..
btw, any good resources here to deal with multiples and divisors?
thanks.
_________________

If my answer helped, dont forget KUDOS!

IMPOSSIBLE IS NOTHING

Math Expert
Joined: 02 Sep 2009
Posts: 27000
Followers: 4169

Kudos [?]: 40036 [0], given: 5397

Re: If M and N are both two digit numbers and M-N=11X, is X an [#permalink]  14 Nov 2012, 07:30
Expert's post
roygush wrote:
If M and N are both two digit numbers and M-N=11X, is X an integer?
(1) The tens digit and units digit of M are the same
(2) M+N is a multiple of 11

I have a problem with this.
Two digit numbers means 13, 24... OR can it also be 22.1 (with decimal/fraction)?

I dont have the OA and thats why im asking..
btw, any good resources here to deal with multiples and divisors?
thanks.

Merging similar topics.
_________________
Intern
Joined: 07 Mar 2013
Posts: 35
Followers: 0

Kudos [?]: 4 [0], given: 80

Re: If m and n are both two digit numbers and m-n = 11x, is x an [#permalink]  10 Sep 2013, 21:25
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
Joined: 10 Oct 2012
Posts: 629
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Kudos [?]: 700 [0], given: 135

Re: If m and n are both two digit numbers and m-n = 11x, is x an [#permalink]  10 Sep 2013, 23:17
Expert's post
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.
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Joined: 06 Sep 2013
Posts: 2023
Concentration: Finance
GMAT 1: 710 Q48 V39
Followers: 24

Kudos [?]: 289 [0], given: 354

Re: If m and n are both two digit numbers and m-n = 11x, is x an [#permalink]  16 Feb 2014, 08:52
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
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Kudos [?]: 2 [0], given: 2

Re: If m and n are both two digit numbers and m-n = 11x, is x an [#permalink]  11 Apr 2015, 19:10
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
Joined: 02 Sep 2009
Posts: 27000
Followers: 4169

Kudos [?]: 40036 [0], given: 5397

Re: If m and n are both two digit numbers and m-n = 11x, is x an [#permalink]  12 Apr 2015, 03:30
Expert's post
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.
_________________
Re: If m and n are both two digit numbers and m-n = 11x, is x an   [#permalink] 12 Apr 2015, 03:30
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