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

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Senior Manager
Joined: 11 Feb 2007
Posts: 340
What is the remainder when the positive integer x is divided  [#permalink]

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Updated on: 06 Sep 2014, 09:04
3
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Difficulty:

95% (hard)

Question Stats:

51% (01:25) correct 49% (01:24) wrong based on 370 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.

When a number must satisfy two different divisible and remainder conditions, you could use what is known as "the Chinese remainder theorem" that uses modular arithmetic. Does anyone know how to apply that theorem to solve this problem?

Or how would you guys solve this in 2 min?

(I picked numbers )

Originally posted by ricokevin on 15 Apr 2007, 05:57.
Last edited by Bunuel on 06 Sep 2014, 09:04, edited 2 times in total.
Edited the question and added the OA.
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Re: integer ??  [#permalink]

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20 Apr 2009, 04:16
1
8
tenaman10 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.

In many remainders questions, it's enough just to find a couple of numbers that 'work' with the given information, and if you simply list the first few numbers that satisfy each statement, it's easy to judge if the statements together are sufficient:

1) x = 5, 17, 29, 41, 53, 65, 77, 89 ...
2) x = 11, 29, 47, 65, 83, 101 ...

Since 29 and 65 give different remainders when you divide by 8, the answer is E.

More abstractly, when combining two statements like the above, the pattern will be based on the LCM of the two divisors. Here, we can consider dividing x by 36 = LCM(12, 18).

Notice that, if Statement 1 is true, the remainder will be 5, 17 or 29 when x is divided by 36.
If Statement 2 is true, the remainder will be 11 or 29 when x is divided by 36.
If both Statements are true, the remainder therefore must be 29 when x is divided by 36.

That's still not sufficient, as above; x could be 29, or x could be 65.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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15 Apr 2007, 08:24
1
Picking numbers is a great way to go!

Clearly, neither is sufficient

We know that x=12k+5 so that x â‚¬ {5,17,29,42,53,65...}

Also x=18m+11 so that xâ‚¬ {11,29,47,65...}

29 and 65, both possible values of x, yield different remainders
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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15 Apr 2007, 09:10
Ok guys this E..

here is why..remember PRIMES...

x=8M + R? where M and R are some integers...

1) x=12M+5; well..12 has 2^2 and 3... 8 has 2^3 so..we dont have enuff 2s to make any decision.. INSUFF

2) x=18M+11; well again look at the primes, 18 has 2 and 3^2; again not enuff 2s ..so Insuff..

together..INSUFF..we still dont have enuff 2s..we need a term that has at least 3 2s ..
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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18 Apr 2007, 12:47
OK, in order to know the remainder..we need to make sure that the statement provide us enuff info to compare with the question stem..

we are asked with is the remainder when X is divided by 8..well 8 has 3 2s..right!

OK statement 1) says X divided by 12 remainder is 5..well...we know that 12 has 2 2s..which is not the same as having 3 2s...so this statement is going to be insuff..

OK statement 2) says x divided by 18, remainder is 11...well 18 again has only 1 2 as a prime factor..again we need some number that has 3 2s...to conclusively anything..so Insuff

combined..again we dont know if the X has 3 2s or not..insuff..

shahrukh wrote:
hey fresinha,
could u explain ur explanation in detail? I dont know the rule of primes in this case.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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05 Sep 2014, 23:37
ricokevin 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.

OA: E

When a number must satisfy two different divisible and remainder conditions, you could use what is known as "the Chinese remainder theorem" that uses modular arithmetic. Does anyone know how to apply that theorem to solve this problem?

Or how would you guys solve this in 2 min?

(I picked numbers )

number plugging -->

1. x = k12+5

k = 0,1,2,3,4.....
x= 5, 17,29,41,53,65...

now x/ 8 --> 5,1,5,1,.....
Not suff

2. x=k18+11
k = 0,1,2,3,4.....
x=11,29,47,65...
nox x/8 --> 3, 5 ,7, 1.....

Not suff..

1+2...

from both we got 29 & 65 are common but still reminders are different.

29 gives rem 5 whereas 65 gives rem 1.

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

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08 Mar 2016, 19:30
Where are the answer options?

ricokevin 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.

When a number must satisfy two different divisible and remainder conditions, you could use what is known as "the Chinese remainder theorem" that uses modular arithmetic. Does anyone know how to apply that theorem to solve this problem?

Or how would you guys solve this in 2 min?

(I picked numbers )
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Joined: 02 Sep 2009
Posts: 49303
Re: What is the remainder when the positive integer x is divided  [#permalink]

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08 Mar 2016, 23:22
abicool456 wrote:
Where are the answer options?

ricokevin 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.

When a number must satisfy two different divisible and remainder conditions, you could use what is known as "the Chinese remainder theorem" that uses modular arithmetic. Does anyone know how to apply that theorem to solve this problem?

Or how would you guys solve this in 2 min?

(I picked numbers )

Hi, and welcome to GMAT Club.

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following post ALL YOU NEED FOR QUANT.

Hope this helps.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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14 Mar 2016, 15:12
Thanks for the explanation! I was wondering if there is any short cut to solve these types of questions? It took me a while to compile all the numbers + I could have easily missed 65 as it is down the line...

luckyme17187 wrote:
ricokevin 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.

OA: E

When a number must satisfy two different divisible and remainder conditions, you could use what is known as "the Chinese remainder theorem" that uses modular arithmetic. Does anyone know how to apply that theorem to solve this problem?

Or how would you guys solve this in 2 min?

(I picked numbers )

number plugging -->

1. x = k12+5

k = 0,1,2,3,4.....
x= 5, 17,29,41,53,65...

now x/ 8 --> 5,1,5,1,.....
Not suff

2. x=k18+11
k = 0,1,2,3,4.....
x=11,29,47,65...
nox x/8 --> 3, 5 ,7, 1.....

Not suff..

1+2...

from both we got 29 & 65 are common but still reminders are different.

29 gives rem 5 whereas 65 gives rem 1.

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

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22 Apr 2016, 07:20
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calinerie wrote:
Thanks for the explanation! I was wondering if there is any short cut to solve these types of questions? It took me a while to compile all the numbers + I could have easily missed 65 as it is down the line...

I did it this way:

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

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23 May 2016, 09:38
1
Icecream87 wrote:
calinerie wrote:
Thanks for the explanation! I was wondering if there is any short cut to solve these types of questions? It took me a while to compile all the numbers + I could have easily missed 65 as it is down the line...

I did it this way:

(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.

What happens when they are equal?

My understanding is that if we add both the equation it will be 0. And if we just take "18b + 11 = x" then we can various values. So insufficient.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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10 Jun 2016, 11:55
ricokevin 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.

Stmt-1:
x= 12a+5
we don’t know the value of ‘a’ so we can’t say whether 12a is divisible by 8. Insuff.

Stmt-2:
x= 18b+11
we don’t know the value of ‘b’ so we can’t say whether 18b is divisible by 8. Insuff

Stmt-1 + Stmt-2:
x= 12a+5
x= 18b+11

Adding both equations we get:

2x = 12a + 18b + 16 now reduce by 2

X = (6a+9b) + 8 , now if we divide by 2 then we are unsure whether (6a+9b) is completely divisible or not. Insufficient.

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

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