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# If j and k are positive integers, j - 2 is divisible by 4

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If j and k are positive integers, j - 2 is divisible by 4 [#permalink]  17 Apr 2011, 10:43
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If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is divisible by 4, all of the following could be the value of j - k EXCEPT:

A. 43
B. 33
C. 21
D. 13
E. 5

[Reveal] Spoiler:
j -2 = 4a => j = 4a + 2
k - 5 = 4b => k = 4b + 5

j - k = 4(a - b) - 3
This means j - k is 3 less than the multiple of 4. That means add 3 to make it the multiple of 4.

43 + 3 = 46 Not the multiple of 4 ---> Answer
33 + 3 = 36. Multiple of 4
21 + 3 = 24 Multiple of 4
13 + 3 = 16 Multiple of 4
5 + 3 = 8 Multiple of 4

Since there is a reversal in thinking - our answer is A.
[Reveal] Spoiler: OA

Last edited by Bunuel on 28 Aug 2012, 05:08, edited 1 time in total.
Edited the question.
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Re: Difficult [#permalink]  17 Apr 2011, 18:37
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j-2 = 4k

=> j = 4k + 2 = 2,6,8,14,18,22,26,30,34,38, 42, 46,50

k-5 = 4j

=> k = 4j + 5 = 5,9,13,17,21,25,29,33

5 = 22 - 17

13 = 30 - 17

21 = 42 - 21

33 = 50 - 17

So by POE, we can see that all the answer choices are posible here except A

We can stop as soon as we see that all other options are possible, by proceeding from lowest value to higher ones.

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Kudos [?]: 75 [1] , given: 42

Re: Difficult [#permalink]  17 Apr 2011, 18:59
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j = 2 +4 (p)

k = 5 + 4(q)

=> j-k = -3+4(...)

A is the only choice that doesnt have this pattern. (i.e multiple of 4 - 3)/

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Re: Difficult [#permalink]  17 Apr 2011, 19:45
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if j -2 is divisible by 4, we can say that j is 4x +2

if k-5 is divisible by 4, it means so will be k-1. and therefore k can be expressed as 4y +1

j - k will be 4x +2- (4y+1) = 4x-4y +2-1 = 4(x-y) +1

The number will have to be a multiple of 4 added to 1.
42 cant be expressed like that . Rest all options can be - (4*8+1, 4*5+1, 4*3+1, 4*1+1)
Hence A.
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Re: Difficult [#permalink]  17 Apr 2011, 20:30
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if j-2 is divisible by 4 and k-5 is divisible by 4 then j-2 - (k-5) is divisible by 4
--> j-k+1 is divisible by 4.

now substitute each of the answer choices in (j-k) + 1 --> 43-1 = 42 --> NOT divisible by 4

Ans A.
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Re: Difficult [#permalink]  18 Apr 2011, 22:04
dreambeliever wrote:
if j-2 is divisible by 4 and k-5 is divisible by 4 then j-2 - (k-5) is divisible by 4
--> j-k+1 is divisible by 4.

now substitute each of the answer choices in (j-k) + 1 --> 43-1 = 42 --> NOT divisible by 4

Ans A.

I may be missing something, how did you get j-k+1?
Should it be j - k + 3 instead?

Ans remains the same. 43+3 is not divisible by 4.
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Re: Difficult [#permalink]  19 Apr 2011, 09:28
Yeah I typed that wrong. It should be j-k+3.

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Re: Difficult [#permalink]  30 Apr 2011, 21:10
j-k = (4a-4b) - 3.
4(a-b) !=46.

Hence A.
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Re: Difficult [#permalink]  28 Aug 2012, 05:04
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Logical way to solve the problem
j-2 = 4a ==> j=4a+2
k-5 = 4b ==> k=4b+5
j-k = 4(a-b) - 3
As the remainder can not be negative we must add divisor to the remainder to make remainder positive
j-k = 4(a-b) + (4-3)
j-k = 4(a-b) +1
i.e. when (j-k) is divided by 4, it will a remainder of 1
All except option "43" leave remainder 1
Solution A
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Re: If j and k are positive integers, j - 2 is divisible by 4 [#permalink]  03 Dec 2012, 01:24
1. If j - 2 is divisible by 4 it can be represented as 4j+2

k - 5 is divisible by 4 and is equal to 4k+5

Thus, j - k is equal to 4j+2-(4k+5)=4(j+k) -3

Which means that the difference of j and k will be a multiple of 4 minus 3.

From there all we need is just to which numbers are multiples of 4.

43+3=46 and it's not divisible by 4
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Re: If j and k are positive integers, j - 2 is divisible by 4 [#permalink]  03 Feb 2014, 02:13
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Re: If j and k are positive integers, j - 2 is divisible by 4 [#permalink]  11 Mar 2014, 00:02
gmat1220 wrote:
If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is divisible by 4, all of the following could be the value of j - k EXCEPT:

A. 43
B. 33
C. 21
D. 13
E. 5

[Reveal] Spoiler:
j -2 = 4a => j = 4a + 2
k - 5 = 4b => k = 4b + 5

j - k = 4(a - b) - 3
This means j - k is 3 less than the multiple of 4. That means add 3 to make it the multiple of 4.

43 + 3 = 46 Not the multiple of 4 ---> Answer
33 + 3 = 36. Multiple of 4
21 + 3 = 24 Multiple of 4
13 + 3 = 16 Multiple of 4
5 + 3 = 8 Multiple of 4

Since there is a reversal in thinking - our answer is A.

j-2 is divisible by 4 & k-5 is divisible by 4, means

j-2 - (k-5) is also divisibly by 4

(j - k + 3) is also divisible by 4

Just putting in the values, 43 is the exception

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Re: If j and k are positive integers, j - 2 is divisible by 4 [#permalink]  23 Jun 2015, 15:08
Hello from the GMAT Club BumpBot!

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If j and k are positive integers, j - 2 is divisible by 4 [#permalink]  27 Jun 2015, 08:49
$$j = 4q + 2$$
$$k = 4q + 5$$

$$j - k = 4q + 2 - (4q + 5) = 2 - 5 = -3$$
Add 4 to -3 to arrive at the remainder value, and therefore R1.

Subtract one from answer choice to calculate whether it's divisible by 4. If not divisible by 4, it meets the criteria.
If j and k are positive integers, j - 2 is divisible by 4   [#permalink] 27 Jun 2015, 08:49
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