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# What is the remainder when the positive integer x is divided

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What is the remainder when the positive integer x is divided  [#permalink]

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Updated on: 07 Oct 2014, 07:18
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Difficulty:

85% (hard)

Question Stats:

55% (01:46) correct 45% (01:43) wrong based on 279 sessions

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What is the remainder when the positive integer x is divided by 8?

(1) When x is divided by 12, the remainder is 5.
(2) When x is divided by 18, the remainder is 11.

Originally posted by albany09 on 07 Oct 2008, 06:25.
Last edited by Bunuel on 07 Oct 2014, 07:18, edited 1 time in total.
Edited the question and added the OA.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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25 Sep 2011, 04:54
4
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albany09 wrote:
What is the remainder when the positive integer x is divided by 8?
(1) When x is divided by 12, the remainder is 5.
(2) When x is divided by 18, the remainder is 11.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

There is a very clean approach to solve such questions within a minute and without writing down anything. But for that you need to understand 'Divisibility' well. If you are willing to do that, read these 4 posts first:
http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/
http://www.veritasprep.com/blog/2011/05 ... emainders/
http://www.veritasprep.com/blog/2011/05 ... s-part-ii/

Now let's look at the question:

What is the remainder when the positive integer x is divided by 8?
This means: What is leftover when you make groups of 8?

Statement 1: When x is divided by 12, the remainder is 5.

When you make groups of 12, 5 balls are leftover. When you make groups of 8 instead, each of the groups of 12 balls leaves 4 balls. If no. of groups of 12 is even, you can combine 2 groups of 4 balls each to make more groups of 8. In that case, 5 balls will be still leftover. So a remainder of 5 is possible.

If no. of groups of 12 is odd, 4 balls will be leftover from one group of 12 and 5 balls will be still leftover. So a total of 9 balls will be leftover. We can make another group of 8 out of these 9 balls and 1 ball will be leftover. So a remainder of 1 is also possible.

Since remainder can be 5 or 1, this statement alone is not sufficient.

Statement 2: When x is divided by 18, the remainder is 11.

When you make groups of 18, 11 balls are leftover. When you make groups of 8 instead, each of the groups of 18 balls makes 2 groups of 8 balls each and leaves 2 balls.
Now there are 4 possibilities:
1. We are left with 2 balls + the original 11 remaining balls = 13 balls
When you make another group of 8 from 13, remainder will be 5
2. We are left with 2+2 balls + the original 11 remaining balls = 15 balls
When you make another group of 8 from 15, remainder will be 7
3. We are left with 2+2+2 balls + the original 11 remaining balls = 17 balls
When you make 2 groups of 8 from 17, remainder will be 1
4. We are left with no groups of 2 balls since they all make a complete group of 8. Only the original 11 balls are remaining. When you make a group of 8 from 11, remainder will be 3.

Since remainder can be 5, 7, 1 or 3, this statement alone is not sufficient.

Using both statements, remainder can be either 5 or 1 so they both together are not sufficient.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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07 Oct 2008, 07:05
albany09 wrote:
What is the remainder when the positive integer x is divided by 8?

(1) When x is divided by 12, the remainder is 5.
(2) When x is divided by 18, the remainder is 11.

1: x = 12k + 5
2: x = 18j + 11

1&2: 12k + 5 = 18j + 11
12k - 18j = 6

some integers (such as 29, 65, 101 or so on) fit to the above eq.

29 divided by 8 has 5 as reminder.
65 divided by 5 has 1 as reminder.

so E.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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23 Sep 2011, 19:01
I chose E as well. But, i dont have an approach to solve these kind of problems. Can anyone help ?
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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12 Oct 2014, 06:09
albany09 wrote:
What is the remainder when the positive integer x is divided by 8?

(1) When x is divided by 12, the remainder is 5.
(2) When x is divided by 18, the remainder is 11.

E.

1) x = (12k+5) ; where k is a non-negative integer
(12k+5) mod 8 = (4k+5) mod 8
when k is even, remainder with 8 will be 5
when k is odd, remainder with 8 will be 1
so, insufficient.

2) x = (18p+5) ; where p is a non-negative integer
(18p+5) mod 8 = (2p+5) mod 8
p=0; remainder = 5
p=1; remainder = 7
p=2; remainder = 1
p=3; remainder = 3
p=4; remainder = 5
and then the cycle repeats.
so, insufficient.

(1)+(2) --> 1 & 5 is common in both lists so insufficient again.
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What is the remainder when the positive integer x is divided  [#permalink]

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22 Apr 2016, 06:50
1
Is my way of doing this correct?:

(1) $$12q + 5 = x$$ insuf

(2) $$18b + 11 = x$$ insuf

Together:
$$12q + 5 = 18b +11$$

$$12q - 18b = 6$$

$$6(2q - 3b) = 6$$

This leads us to $$2q - 3b = 1$$

I replace q by b in equation 1:

$$12(\frac{1 + 3b}{2}) +5 = x$$

$$18b + 11 = x$$

Statement 1 becomes equal to statement 2 : Hence insufficient.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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16 Jul 2016, 23:55
x = 8K + r;
1) x = 12P + 5 -> 2 equation, 3 unknown = Can't be solved; insufficient.
2) x = 18Q + 11 -> 2 equation, 3 unknown = Can't be solved; insufficient.

Combine -> 3 equation, 4 unknown = Can't be solved; insufficient.

Option E.

Note: If there can be a way to reduce variable eg. one var is factor of other var, we could have found a solution.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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17 Jul 2016, 02:20
Option 1 : Possible Numbers could be 5,17,29,41,53,65,...

when divided by 8, they will give remainders are 5,1,5,1,...

Hence, not sufficient.

Option 2 : Possible Numbers could be 11,29,47,65,...

when divided by 8, they will give remainders are 3,5,7,5,...

Hence, not sufficient.

Combining both the statements, Possible no. could be 29 and 65.

When divided by 8,They will give remainders as 5,1.

Hence, after combining also, it is insufficient. Correct Answer : E.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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08 Mar 2017, 12:57
My way of doing

x = 8k + r, r = ?

S-1) x = 12k + 5, Possible values x = 5, 17,29. Not sufficient
S-2) x = 18k + 11, x = 11,29. Not sufficient
S-T) Using above 2 statements and LCM of 12 & 18 we get 36 and first common term for the two statements is 29,
x = 36k+29, x = 29, 65. So remainders are 5 or 1. NS.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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14 Sep 2017, 09:33
x/8 => remainder =?
1) x/12 => R=5
x= 5,17,29,41,53,65,77,89,101...
R= 5,1 ( A,D eliminated)

2) x/18=> R=11
x= 11,47,65,_,_,101,....
R= 3,7 ( B eliminated)

Combine

x= 65,101,...
R= 1,5 ( C eliminated)

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Re: What is the remainder when the positive integer x is divided &nbs [#permalink] 14 Sep 2017, 09:33
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