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If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is

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Bunuel
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If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is [#permalink] New post   12 Nov 2019, 02:06
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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


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A
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Re: If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is [#permalink] New post   12 Nov 2019, 03:11
from given info
we can say
j-2-k+5/4
j-k+3/4
plug in values for j-k from given options
only option A ; 43 is not divisible
IMO A 43


Bunuel 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


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Re: If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is [#permalink] New post   12 Nov 2019, 05:51
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\(\frac{j - 2}{4} \)= m (where m is an integer) and \(\frac{k - 5}{4}\) = n (where n is an integer) ==> \(\frac{j - 2}{4} \) - \(\frac{k - 5}{4}\) = p (where p is an integer) ==> j - k = 4p - 3 ==> check the answer choices, and the answer that cannot be expressed in this form is the correct answer

A. 43
B. 33 = 4(9) - 3
C. 21 = 4(6) - 3
D. 13 = 4(4) - 3
E. 5 = 4(2) - 3

Therefore, the answer is A.
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Re: If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is [#permalink] New post   12 Nov 2019, 11:13
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J=4p+2
K=4q+5
j-k = 4(p-q)-3
Insert values from options. Correct answer after adding 3 to it will be divisible by 4.
4(p-q)-3=43 ; 46 not divisible by 4
4(p-q)-3=33 ; 36 divisible by 4
4(p-q)-3=21 ; 24 divisible by 4
4(p-q)-3=13 ; 16 divisible by 4
4(p-q)-3=5 ; 8 divisible by 4

Hence, correct answer choice is A.
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Re: If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is [#permalink] New post   12 Nov 2019, 12:06
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j-2=4x
j=4x+2

k-5=4y
k=4(y+1)+1

All the options are positive; hence j>k

j-k= 4(x-y-1)+1
j-k leaves remainder 1, when divided by 4

All options except A leaves remainder 1, when divided by 4.

Bunuel 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


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Re: If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is [#permalink] New post   20 Nov 2019, 18:52
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Bunuel 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


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Since j - 2 and k - 5 are both divisible by 4, their difference is also divisible by 4; that is:

j - 2 - (k - 5) = j - k + 3 is divisible by 4.

Since j - k + 3 is divisible by 4, then when j - k is divided by 4, it should have a remainder of 1 (notice that if j - k is 1, then j - k + 3 = 4). Therefore, we are looking for the choice that doesn’t yield a remainder of 1 when it is divided by 4, and we see that is the number 43 (all the other choices do indeed yield a remainder of 1 when divided by 4).

Answer: A
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Re: If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is [#permalink] New post   24 Nov 2019, 20:10
Expert Reply

Solution


Given:
    • j and k are positive integers
    • j – 2 is divisible by 4
    • k – 5 is divisible by 4

To find:
    • Which cannot be the value of j - k

Approach and Working Out:
    • From the given information we can write,
      o \(j – 2 = 4p_1\), where \(p_1\) is an integer
         Implies, \(j = 4p_1 + 2\)
      o Similarly, \(k – 5 = 4p_2\), where, \(p_2\) is an integer
         \(k = 4p_2 + 5\)
    • Thus, \(j – k = 4p_1 + 2 – 4p_2 – 5 = 4(p_1 – p_2) – 3 = 4(p_1 – p_2) – 4 + 1 = 4(p_1 – p_2 – 1) + 1= 4p + 1\)
    • The only option which is not in the form of 4p + 1 is option A, 43

Hence, the correct answer is Option A.

Answer: A
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Re: If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is [#permalink] New post   22 Jul 2023, 04:54
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Re: If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is [#permalink]
22 Jul 2023, 04:54
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