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

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

Let's say N = 6k +1
When N is ODD , Values of N are 7,19,31,43...
The remainder when N is divided by 12 is 7.

When N is even, Values of N are 1,13,24,37..
The remainder when N is divided by 12 is 1.

So we can conclude the when N is divided by 12 , the remainder could be 1 or 7.
Hence Statement 1 alone is insufficient.

Statement 2:When n is divided by 12, the remainder is greater than 5.
So the remainder could be 6,7,8,9,10,11
So Statement 2 alone is insufficient.

When you combine both statements, we can conclude that the Remainder is 7
option C is the right answer.

Thanks,
Clifin J Francis
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
Statement 1 is clearly insufficient, coz N could be 1,7,13,19,25,31 and each one of those possible values of N gives different value for the remainder when divided by 12

But i donot understand statement 2.

This is what i get. if N/12 and the remainder is greater than 5, this means N could be 6,7,8,9,10,11 but N could also be 18,19,20,21,22,23.

If N 18, then 18/12, the remainder is 6,which is greater than 5

If N 19, then 19/12, the remainder is 7,which is greater than 5

etc....

Conclusion/

From statement 1 N could be 1,7,13,19,25,31

From statement 2 N could be 6,7,8,9,10,11 but N could also be 18,19,20,21,22,23

Both statement together: N could 7 or 19 and that is why i have E as answer choice.

Can anyone tell me where i go wrong in my approach?

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

avigutman sir I am not able to approach the 2 statements