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# M10-12

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Math Expert
Joined: 02 Sep 2009
Posts: 58320
M10-12  [#permalink]

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16 Sep 2014, 00:41
1
8
00:00

Difficulty:

15% (low)

Question Stats:

82% (01:27) correct 18% (01:47) wrong based on 91 sessions

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If $$X$$ is a positive integer, what is the remainder of $$\frac{X}{8}$$?

(1) The remainder of $$\frac{X}{16}$$ is 2.

(2) The remainder of $$\frac{X}{24}$$ is 10.

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Re M10-12  [#permalink]

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16 Sep 2014, 00:41
1
1
Official Solution:

Statement (1) by itself is sufficient. From S1 it follows that $$X = 16N + 2$$ where $$N$$ is an integer. Thus, the remainder of $$\frac{X}{8}$$ is 2.

Statement (2) by itself is sufficient. From S2 it follows that $$X = 24N + 10 = (24N + 8) + 2$$ where $$N$$ is an integer. Thus, the remainder of $$\frac{X}{8}$$ is 2.

Answer: D
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Re: M10-12  [#permalink]

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09 Nov 2014, 08:07
I'd appreciate if you could explain what you did for st 2.. how did you get x/8 ?
Manager
Joined: 07 Dec 2009
Posts: 86
GMAT Date: 12-03-2014
Re: M10-12  [#permalink]

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10 Dec 2014, 15:24
1
1
Split 24N + 10 ==> (24N+8) + 2

It can further be written as 8(3N+1) + 2. When this is divided by 8 the remainder is 2.
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Joined: 27 Aug 2014
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Re: M10-12  [#permalink]

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23 Feb 2016, 00:48
Bunuel wrote:
Official Solution:

Statement (1) by itself is sufficient. From S1 it follows that $$X = 16N + 2$$ where $$N$$ is an integer. Thus, the remainder of $$\frac{X}{8}$$ is 2.

Statement (2) by itself is sufficient. From S2 it follows that $$X = 24N + 10 = (24N + 8) + 2$$ where $$N$$ is an integer. Thus, the remainder of $$\frac{X}{8}$$ is 2.

Answer: D

Hi

I think I am missing some concept here. I framed equations like this:
Stem=X=8a+R; To find R=X-8a
Stmt1=X=16b+2
Stmt2=X=24c+10

Hence got E as cant find a or X from any of the stmts. I knw logically what I did is not wrong but dont know how to proceed from here. Read something about primes but how does the concept come about? Pls elaborate. Thanks.
Math Expert
Joined: 02 Aug 2009
Posts: 7953
M10-12  [#permalink]

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23 Feb 2016, 01:01
3
3
sinhap07 wrote:
Bunuel wrote:
Official Solution:

Statement (1) by itself is sufficient. From S1 it follows that $$X = 16N + 2$$ where $$N$$ is an integer. Thus, the remainder of $$\frac{X}{8}$$ is 2.

Statement (2) by itself is sufficient. From S2 it follows that $$X = 24N + 10 = (24N + 8) + 2$$ where $$N$$ is an integer. Thus, the remainder of $$\frac{X}{8}$$ is 2.

Answer: D

Hi

I think I am missing some concept here. I framed equations like this:
Stem=X=8a+R; To find R=X-8a
Stmt1=X=16b+2
Stmt2=X=24c+10

Hence got E as cant find a or X from any of the stmts. I knw logically what I did is not wrong but dont know how to proceed from here. Read something about primes but how does the concept come about? Pls elaborate. Thanks.

Hi,
NOTE:-
whatever is the remainder by an integer, the remainder will remain teh same if div by its factors..
only thing to check is if the R is further div by that factor

what you should have done after making your statements..
Stmt1=X=16b+2
we do not require X or a, we are interested in R..
so what does x/8 mean..
(16b+2)/8=16b/8 + 2/8..
16b is div by 8 so will not leave any remainder but 2/8 will leave a remainder of 2..
suff

Stmt2=X=24c+10

same way ( 24c+10)/8= 3c +10/8..
when 10 is div by 8, it will give us 2 as remainder ..
again suff

D

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Re: M10-12  [#permalink]

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03 May 2016, 06:09
could someone explain to me how does it follow from statement 1 that

that X=16N+2X=16N+2 where N is an integer?
Math Expert
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Re: M10-12  [#permalink]

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03 May 2016, 06:22
nausherwan wrote:
could someone explain to me how does it follow from statement 1 that

that X=16N+2X=16N+2 where N is an integer?

hi
I. (1) The remainder of $$\frac{X}{16}$$ is 2...
it can be written as -- when x is divided by 16, the remainder is 2...
we do not know the quotient(Q) here but just know the remainder..
how do I write it in a equation-
x = 16a + 2..
a is the quotient and 2 is the remainder..
say a is 5.. so x= 5*16 +2 = 82..
I can write it as-- the remainder of 82/16 is 2..

so x=16n+2 is just another way of writing -- The remainder of $$\frac{X}{16}$$ is 2
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Re: M10-12  [#permalink]

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03 May 2016, 06:24
1
nausherwan wrote:
could someone explain to me how does it follow from statement 1 that

that X=16N+2X=16N+2 where N is an integer?

Check the links below:
Divisibility and Remainders on the GMAT

Theory on remainders problems

Tips on remainders

Cyclicity on the GMAT

Units digits, exponents, remainders problems

Hope it helps.
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Re: M10-12  [#permalink]

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14 Nov 2016, 22:58
why did you consider N as an integer.it was not given in the question?
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Joined: 12 Jun 2016
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Re: M10-12  [#permalink]

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27 May 2017, 10:08
Another approach - Write out few of the Numbers and check

S1: The Remainder of $$\frac{X}{16}= 2$$
This means X = 2, 18, 34...
Remainders when divided by 8 in all the cases = 2
Suff

S2: The Remainder of $$\frac{X}{24}= 10$$
This means X = 10, 34, 58...
Remainder when X is divided by 8 in all cases = 2
Suff

Final answer : D
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Re M10-12  [#permalink]

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31 Jan 2018, 00:53
I think this is a high-quality question and I don't agree with the explanation. what if we take x=14 for statement 1.
then 14/16 gives rem as 2
but 14/8 gives rem as 6.
so the correct answer should be B.
Math Expert
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Re: M10-12  [#permalink]

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31 Jan 2018, 01:49
muditsingh0616 wrote:
I think this is a high-quality question and I don't agree with the explanation. what if we take x=14 for statement 1.
then 14/16 gives rem as 2
but 14/8 gives rem as 6.
so the correct answer should be B.

14 divided by 16 gives the remainder of 14, not 2. From (1) x could be 2, 18, 34, 50, 66, ... Each of these values gives a remainder of 2 when divided by 8, so (1) is sufficient.
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Re: M10-12  [#permalink]

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04 Nov 2018, 11:05
Don't see how S) 1 is sufficient.

x/16 gives remainder 2.

Hence X = 16q + 2

if q=0,
x= 16(0) + 2
=2

if q=1
x= 16 + 2
= 18

lets take x=2. 2/8 gives 0 remainder 8.
lets take x=18 18/8 gives 2 remainder 2.

Insufficient.

S) 2 on the other hand..

X = 24p + 10

if p=0
x= 10

p=1
x = 34

10/8 gives remainder 2
34/8 gives remainder 2.

Statement 2 works, but 1 doesn't.
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Re: M10-12  [#permalink]

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08 Mar 2019, 13:59
PierTotti17 wrote:
Don't see how S) 1 is sufficient.

x/16 gives remainder 2.

Hence X = 16q + 2

if q=0,
x= 16(0) + 2
=2

if q=1
x= 16 + 2
= 18

lets take x=2. 2/8 gives 0 remainder 8.
lets take x=18 18/8 gives 2 remainder 2.

Insufficient.

S) 2 on the other hand..

X = 24p + 10

if p=0
x= 10

p=1
x = 34

10/8 gives remainder 2
34/8 gives remainder 2.

Statement 2 works, but 1 doesn't.

PierTotti17 All is good in what you wrote, except, 2/8 gives a remainder of 2, not 8. Hence 1 is sufficient as well.
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Re: M10-12  [#permalink]

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25 Aug 2019, 08:12
Easy to prove sufficiency with number picking, just plug in 2, 18 for St 1) you will get remainder 2 both times when dividing by 8.
St 2) must give the same remainder... you get 2 when dividing by 8 as well, plug in 2 and 34.
This is because both 16 and 24 are multiples of 8 so they all share the same remainder when dividing X. As chetan2u mentioned the only thing is to make sure remainder/divisor is in the most reduced form.
Re: M10-12   [#permalink] 25 Aug 2019, 08:12
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