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If positive integer x is divided by 5, the result is p and [#permalink]
08 Apr 2012, 02:33

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Question Stats:

75% (02:26) correct
25% (01:05) wrong based on 159 sessions

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 [#permalink]
08 Apr 2012, 02:47

4

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Expert's post

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.

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.

Re: If positive integer x is divided by 5, the result is p and [#permalink]
20 Apr 2012, 09:26

1

This post received KUDOS

Expert's post

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

Re: If positive integer x is divided by 5, the result is p and [#permalink]
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.

Re: If positive integer x is divided by 5, the result is p and [#permalink]
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.

Re: If positive integer x is divided by 5, the result is p and [#permalink]
07 Mar 2013, 20:02

Expert's post

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. _________________

Re: If positive integer x is divided by 5, the result is p and [#permalink]
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 _________________

......................................................................... +1 Kudos please, if you like my post

Re: If positive integer x is divided by 5, the result is p and [#permalink]
24 Aug 2015, 03:36

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