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What is the remainder when the positive integer n is divided by 12?  [#permalink]

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What is the remainder when the positive integer n is divided by 12?

(1) When n is divided by 6, the remainder is 1.
(2) When n is divided by 12, the remainder is greater than 5.

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Originally posted by windofchange on 07 Oct 2013, 03:05.
Last edited by Bunuel on 29 Oct 2017, 00:32, edited 2 times in total.
Renamed the topic and edited the question.
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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What is the remainder when the positive integer n is divided by 12?

The remainder is always non-negative integer less than divisor $$0\leq{r}<d$$, so in our case $$0\leq{r}<12$$.

(1) When n is divided by 6, the remainder is 1 --> $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25, ... This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.

(2) When n is divided by 12, the remainder is greater than 5. This implies that $$5<{r}<12$$. Not sufficient.

(1)+(2) Since from (2) $$5<{r}<12$$, the from (1) r=7. Sufficient.

Hope it's clear.
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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Bunuel wrote:
What is the remainder when the positive integer n is divided by 12?

The remainder is always non-negative integer less than divisor $$0\leq{r}<d$$, so in our case $$0\leq{r}<12$$.

(1) When n is divided by 6, the remainder is 1 --> $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25, ... This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.

(2) When n is divided by 12, the remainder is greater than 5. This implies that $$5\leq{r}<12$$. Not sufficient.

(1)+(2) Since from (2) $$5\leq{r}<12$$, the from (1) r=7. Sufficient.

Hope it's clear.

Hi Bunuel, can you post some practice problems for the 'Remainders' topic? I got this in my GMAT Prep exam and I got it wrong. I'd like to review this topic a bit more. Thanks.
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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saintforlife wrote:
Bunuel wrote:
What is the remainder when the positive integer n is divided by 12?

The remainder is always non-negative integer less than divisor $$0\leq{r}<d$$, so in our case $$0\leq{r}<12$$.

(1) When n is divided by 6, the remainder is 1 --> $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25, ... This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.

(2) When n is divided by 12, the remainder is greater than 5. This implies that $$5\leq{r}<12$$. Not sufficient.

(1)+(2) Since from (2) $$5\leq{r}<12$$, the from (1) r=7. Sufficient.

Hope it's clear.

Hi Bunuel, can you post some practice problems for the 'Remainders' topic? I got this in my GMAT Prep exam and I got it wrong. I'd like to review this topic a bit more. Thanks.

Theory on remainders problems: remainders-144665.html

All DS remainders problems to practice: search.php?search_id=tag&tag_id=198
All PS remainders problems to practice: search.php?search_id=tag&tag_id=199

Hope this helps.
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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Hi Bunuel,

Maybe I'm just rusty on remainder theory, but you would please explain how you were able to see this:

Quote:
This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.

Thanks,
MDL
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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mdlyman wrote:
Hi Bunuel,

Maybe I'm just rusty on remainder theory, but you would please explain how you were able to see this:

Quote:
This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.

Thanks,
MDL

1, 7, 13, 19, 25,

1 divided by 12 gives the remainder of 1;
7 divided by 12 gives the remainder of 7;
13 divided by 12 gives the remainder of 1;
19 divided by 12 gives the remainder of 7;
25 divided by 12 gives the remainder of 1;
...

Check links for theory on remainders in my post above.

Hope it helps.
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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Bunuel wrote:
What is the remainder when the positive integer n is divided by 12?

The remainder is always non-negative integer less than divisor $$0\leq{r}<d$$, so in our case $$0\leq{r}<12$$.

(1) When n is divided by 6, the remainder is 1 --> $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25, ... This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.

(2) When n is divided by 12, the remainder is greater than 5. This implies that $$5\leq{r}<12$$. Not sufficient.

(1)+(2) Since from (2) $$5\leq{r}<12$$, the from (1) r=7. Sufficient.

Hope it's clear.

I have a doubt here , like u said $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25 so we are substituting q with 0,1 ,2,3 ... so on ... But I wanted to know if we can substitude 0 . .... That means if i divide 1/6 ---is the remainder 1 . But here I cannot divide in the first place only. Please clear my concept , i guess i m missing something . I thought we can only get remainder when the no is atleast divisible once , means p/q where p > q .... thanks in advance
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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hanschris5 wrote:
Bunuel wrote:
What is the remainder when the positive integer n is divided by 12?

The remainder is always non-negative integer less than divisor $$0\leq{r}<d$$, so in our case $$0\leq{r}<12$$.

(1) When n is divided by 6, the remainder is 1 --> $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25, ... This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.

(2) When n is divided by 12, the remainder is greater than 5. This implies that $$5\leq{r}<12$$. Not sufficient.

(1)+(2) Since from (2) $$5\leq{r}<12$$, the from (1) r=7. Sufficient.

Hope it's clear.

I have a doubt here , like u said $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25 so we are substituting q with 0,1 ,2,3 ... so on ... But I wanted to know if we can substitude 0 . .... That means if i divide 1/6 ---is the remainder 1 . But here I cannot divide in the first place only. Please clear my concept , i guess i m missing something . I thought we can only get remainder when the no is atleast divisible once , means p/q where p > q .... thanks in advance

Let me ask you a question: how many leftover apples would you have if you had 1 apple and wanted to distribute in 6 baskets evenly? Each basket would get 0 apples and 1 apple would be leftover (remainder).

When a divisor is more than dividend, then the remainder equals to the dividend, for example:
3 divided by 4 yields the reminder of 3: $$3=4*0+3$$;
9 divided by 14 yields the reminder of 9: $$9=14*0+9$$;
1 divided by 9 yields the reminder of 1: $$1=9*0+1$$.

Theory on remainders problems: remainders-144665.html

All DS remainders problems to practice: search.php?search_id=tag&tag_id=198
All PS remainders problems to practice: search.php?search_id=tag&tag_id=199

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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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naeln wrote:
What is the remainder when the positive integer n is divided by 12?
1. When n is divided by 6, the remainder is 1
2. When n is divided by 12, the remainder is greater than 5

The question is asking what would the remainder be when n is divided by 12. We know that remainder obtained when n is divided by p is < p. Thus remainders when any integer is divided is divded by 12 will be one of 0,1,2,3,4,5,6,7,8,9,10,11.

Per statement 1, n =6p+1 --> n = 7 (remainder when divided by 12 = 7), or n =13 (remainder when divided by 12 = 1). Thus we get 2 different values for the remainder. Not suficient.

Per statement 2, n =12q+ r where r >5 ---> r could be one of 6-11. Thus not sufficient.

Combining, we get that the remainder will be either 1 or 7 and that the remainder will be >5 . Thus remainder will be 7. C is the correct answer.
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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mdlyman wrote:
Hi Bunuel,

Maybe I'm just rusty on remainder theory, but you would please explain how you were able to see this:

Quote:
This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.

Thanks,
MDL

The best way to think of remainders is using the Number line. We all know that every 3rd number starting at 3 is multiple of 3. Looking at this another way, every 3rd number on the number line starting with 3, yields a remainder of 0 when divided by 3. Similarly, every 3rd number on the number line starting with the number 4, will yield a remainder of 1 when divided by 3.

In the problem above, a number that yields a remainder of 1 when divided by six would be every sixth number on the number line starting a 1 (i.e., 1, 7, 13, 19, .....). Dividing these same numbers by 12, yields remainders of 1 (13) or 7 (19).
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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Instead of testing values, You can just divide the number by 12(or anything for any question) and check for the remainders
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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windofchange wrote:
What is the remainder when the positive integer n is divided by 12?

(1) When n is divided by 6, the remainder is 1.
(2) When n is divided by 12, the remainder is greater than 5.

We need to determine the remainder when n is divided by 12.

Statement One Alone:

When n is divided by 6, the remainder is 1.

The information in statement one is not sufficient to answer the question. We see that when n = 7, 7/12 has a remainder of 7; however when n = 13, 13/12 has a remainder of 1.

Statement Two Alone:

When n is divided by 12, the remainder is greater than 5.

The information in statement two is not sufficient to answer the question, since when n is divided by 12, it can be any one of these possible remainders: 6, 7, 8, 9, 10, and 11.

Statements One and Two Together:

Using the information from statements one, we see that n can be values such as:

7, 13, 19, 25, …..

We also see that when we divide these values by 12, we get a pattern of remainders:

7/12 has a remainder of 7

13/12 has a remainder of 1

19/12 has a reminder of 7

25/12 has a remainder of 1

Since we have found a pattern, we do not have to test any further numbers. Furthermore, since statement two tells us that the remainder when N is divided by 12 is greater than 5, the only possible remainder is 7.

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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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Hi,
Please, I have problem understanding why c. In my opinio, the answer is E because not only 7 meets the criteria but also 31 !
can you tell me where am i wrong ?

Thank you !
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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hichem wrote:
Hi,
Please, I have problem understanding why c. In my opinio, the answer is E because not only 7 meets the criteria but also 31 !
can you tell me where am i wrong ?

Thank you !

The question asks to find the value of r not n. What is the remainder when the positive integer n is divided by 12? That being said, n can take infinitely many values: 7, 19, 31, 43, ... not just the two you mention.
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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windofchange wrote:
What is the remainder when the positive integer n is divided by 12?

(1) When n is divided by 6, the remainder is 1.
(2) When n is divided by 12, the remainder is greater than 5.

the first number to satisfy both conditions is 17
all the number satisfying both condition will be
17+ k. minimum multiple of 6 and 12
=17+ k. 12
so, we find that the remainder is alway 2. C is correct

we need to find the minimum multitude of 6 and 12. which is 12. the question will become more dificult if it is harder to find minimum multiple of 6 and 12
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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We can also use the Dividend and Divisor property.

Statement 1: When n is divided by 6, the remainder is 1.

This means that n = 6 * q + 1 (1 Equation and 2 Variables) [Clearly not sufficient]

Statement 2: When n is divided by 12, the remainder is greater than 5

This means that n = 12 * q + 6 (or the range can be anything) (1 Equation and 2 Variables) [Clearly not sufficient]

Combining Statement 1 + Statement 2 = We can easily solve for both the equations in 2 varibles.

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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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dheerajt94 wrote:
We can also use the Dividend and Divisor property.

Statement 1: When n is divided by 6, the remainder is 1.

This means that n = 6 * q + 1 (1 Equation and 2 Variables) [Clearly not sufficient]

Statement 2: When n is divided by 12, the remainder is greater than 5

This means that n = 12 * q + 6 (or the range can be anything) (1 Equation and 2 Variables) [Clearly not sufficient]

Combining Statement 1 + Statement 2 = We can easily solve for both the equations in 2 varibles.

Hey i may be late but i wanted to point out that your approach is flawed!

Both the "q" values u have taken are same in both equations of "n". But actually there will be two different values!
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Re: What is the remainder when the positive integer n is divided by 12?  [#permalink]

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Here are my two cents for this question

Using Even ODD concept.

division by 12 can yield any reminder from 0 to 11
Now
$$\frac{n}{6}$$reminder is 1 we can write this algebraically as n=6K+1

now when $$\frac{n}{12}$$ =$$\frac{6K+1}{12} w$$e can have two possible out comes reminder If K is even we have 1 as remainder and if K is odd , reminder as 7

Now as per second statement when $$\frac{n}{12}$$ we get a reminder which is greater than 5, so reminder can be any number from 6 to 11
or we can write it allergically as n= 12Q+R

Now combining both we have
( if some one did miss the first inference that reminder can be 1 or 7)
we could see that 12Q+R=6K+1 where 5<R<11

so we have
12Q-6K=1-R
Recognize that
6(2Q-K)= 1-R
LHS is Even So RHS must be even so R must be odd. Possible values of R are 7,9,11

But recognize that LHS is multiple of 6 so should also be RHS . the only value which gives us multiple of 6 on RHS is if the value of R is 7

hence we can say the remainder is 7

Hence we can answer out question using both statements

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