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If positive integer x is divided by 5, the result is p and

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If positive integer x is divided by 5, the result is p and [#permalink] New post 08 Apr 2012, 02:33
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If positive integer x is divided by 5, the result is p and the remainder 3. If x is divided by 11, the remainder is 3 again, what is the remainder when p is divided by 11?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

I had to plug in numbers, how can you solve this with the remainder formula?
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If positive integer x is divided by 5, the result is p and [#permalink] New post 08 Apr 2012, 02:47
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BN1989 wrote:
If positive integer x is divided by 5, the result is p and the remainder 3. If x is divided by 11, the remainder is 3 again, what is the remainder when p is divided by 11?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

I had to plug in numbers, how can you solve this with the remainder formula?


If positive integer x is divided by 5, the result is p and the remainder 3: x=5p+3;
If positive integer x is divided by 11, the the remainder 3: x=11q+3;

Subtract one from another: x-x=(5p+3)-(11q+3) --> 5p=11q---> \frac{p}{q}=\frac{11}{5} --> since both p and q are integers then p must be a multiple of 11, so it yields remainder of zero upon division by 11.

Answer: A.

Hope it's clear.
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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 08 Apr 2012, 04:35
I got this wrong....

I tried to plug in numbers but didn't manage to get it. Then just guessed

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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 19 Apr 2012, 22:35
Bunuel, I tried using this method below as described in

http://gmatclub.com/forum/manhattan-remainder-problem-93752.html#p721341

I got stuck. Please help

X=5P+3 , x can be 8 13 18 23...58
X=11Q+3, x can be 14,25,....58

To form the equation n=kx+r
n=55K+58

Not sure how to proceed.
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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 19 Apr 2012, 23:31
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Since X-3 is divisible both by 5 and by 11,which are prime numbers, so P/11 or X-3/11 will always be with remainder 0
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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 20 Apr 2012, 02:35
Expert's post
ENAFEX wrote:
Bunuel, I tried using this method below as described in

http://gmatclub.com/forum/manhattan-remainder-problem-93752.html#p721341

I got stuck. Please help

X=5P+3 , x can be 8 13 18 23...58
X=11Q+3, x can be 14,25,....58

To form the equation n=kx+r
n=55K+58

Not sure how to proceed.


First of all you don't need to use that approach to solve the problem.

Next, you are making a mistake while deriving a general formula.

Positive integer x is divided by 5, the result is p and the remainder 3: x=5p+3 --> x can be: 3, 8, 13, ... Notice that the least value of x for which it gives the remainder of 3 upon division by 5 is 3 itself: 3 divided by 5 yields remainder of 3.

Positive integer x is divided by 11, the the remainder 3: x=11q+3 --> x can be: 3, 14, 25, ... Th same here the least value of x is 3: 3 divided by 11 yields remainder of 3.

General formula would be: x=55k+3.

Check the problems below for which you can use this approach:
positive-integer-n-leaves-a-remainder-of-4-after-division-by-93752.html
if-n-is-a-positive-integer-greater-than-16-is-n-a-prime-129829.html
when-positive-integer-x-is-divided-by-5-the-remainder-is-128470.html
when-n-is-divided-by-5-the-remainder-is-2-when-n-is-divided-82624.html
when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html
when-the-positive-integer-a-is-divided-by-5-and-125591.html
what-is-the-value-of-length-n-100-meter-of-wire-126500.html
a-group-of-n-students-can-be-divided-into-equal-groups-of-126384.html
when-the-positive-integer-a-is-divided-by-5-and-7-the-104480.html
positive-integer-n-leaves-a-remainder-of-4-after-division-by-93752.html
when-positive-integer-n-is-divided-by-3-the-remainder-is-86155.html

Hope it helps.
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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 20 Apr 2012, 09:26
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BN1989 wrote:
If positive integer x is divided by 5, the result is p and the remainder 3. If x is divided by 11, the remainder is 3 again, what is the remainder when p is divided by 11?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

I had to plug in numbers, how can you solve this with the remainder formula?


If the remainder is same in both the cases,
x = 5p + 3
x = 11q + 3

then x = 55a + 3
(to understand this concept, check out http://www.veritasprep.com/blog/2011/05 ... emainders/)

Since 5p has 55 as a factor, p must be divisible by 11. So remainder is 0
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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 19 Oct 2012, 00:27
x=5p+3;
x=11q+3

So, 5p+3=11q+3..
5p=11q
p=11(q/5)

P should be a multiple of 11... & p divided by 11 should give R=0
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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 27 Nov 2012, 03:35
ENAFEX wrote:
Bunuel, I tried using this method below as described in

I got stuck. Please help

X=5P+3 , x can be 8 13 18 23...58
X=11Q+3, x can be 14,25,....58

To form the equation n=kx+r
n=55K+58

Not sure how to proceed.


Using the same approach, we know that at p=11 the value of X=58, for both the expressions. Hence p is a multiple of 11 so the remainder is 0.
Though this is still a more time consuming approach that the ones stated above.

Please correct me if I am wrong.

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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 07 Mar 2013, 09:23
Hi I have a quick question on this problem. How are you getting to 55 in the combined equation? Why can't X be 3? If you divide 3 by both 5 and 11, the remainder is 3 so I'm not sure what I am missing. Thanks for any help you can give.
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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 07 Mar 2013, 20:02
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aryah422 wrote:
Hi I have a quick question on this problem. How are you getting to 55 in the combined equation? Why can't X be 3? If you divide 3 by both 5 and 11, the remainder is 3 so I'm not sure what I am missing. Thanks for any help you can give.


I have discussed the general case there.

Given that:
x = 5p + 3
x = 11q + 3

We can say that x = 55a + 3
i.e. when we divide x by 55 (the LCM of 5 and 11), the remainder will be 3 in that case too.
To understand this fully, check out the link I gave in my previous post: http://www.veritasprep.com/blog/2011/05 ... emainders/

Sure, the number x can be 3 too. In that case p = 0, q = 0 and a = 0. When you divide p by 11, the remainder will be 0.
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Re: If positive integer x is divided by 5, the result is p and [#permalink] New post 31 Jul 2014, 01:15
BN1989 wrote:
If positive integer x is divided by 5, the result is p and the remainder 3. If x is divided by 11, the remainder is 3 again, what is the remainder when p is divided by 11?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

I had to plug in numbers, how can you solve this with the remainder formula?


When we got 5p = 11k, since 5 and 11 is prime number -> k must be divisible by 5 and p must be divisible by 11 -> A is correct
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Re: If positive integer x is divided by 5, the result is p and   [#permalink] 31 Jul 2014, 01:15
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