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Question of the Week- 10 ( When an even integer p is divided by 7,...)

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Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post Updated on: 12 Aug 2018, 23:14
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e-GMAT Question of the Week #10


When an even integer p is divided by 7, the remainder is 6. Which of the following, when added to p, gives a sum that leaves a remainder of 4 when divided by 14?

Options

(a) 10
(b) 14
(c) 16
(d) 26
(e) 32

To access all the questions: Question of the Week: Consolidated List

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Originally posted by EgmatQuantExpert on 05 Aug 2018, 21:42.
Last edited by EgmatQuantExpert on 12 Aug 2018, 23:14, edited 2 times in total.
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Re: Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post 05 Aug 2018, 22:07
EgmatQuantExpert wrote:
e-GMAT Question of the Week #10


When an even integer p is divided by 7, the remainder is 6. Which of the following, when added to p, gives a sum that leaves a remainder of 4 when divided by 14?

Options

(a) 10
(b) 14
(c) 16
(d) 26
(e) 32

To access all the QOW questions: Question of the Week: Consolidated List


so p = 2n, that is an even number
p=2n=7k+6 so k has to be an even integer
the above basically means that when p is divided by 14 or (7k), the remainder is 6..
and when a number is added, remainder 4 is left..

therefore 6+new number -4 = new number +2 should be div by 14
(a) 10........ 10+2=12... NO
(b) 14........ 14+2=16... NO
(c) 16........ 16+2=18... NO
(d) 26........ 26+2=28.. YES
(e) 32........ 32+2=34... NO

ans D
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Re: Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post 05 Aug 2018, 23:15
1
If an integer leaves remainder 4 with 14,it will leave the same remainder when divided by 7.
So,we need to find a number that leaves rem 4 when div by 7
A.7p+6+10=>rem. 2(when div by 7)
B.7p+6+14=>rem. 6
C.7p+6+16=>rem. 1
D.7p+6+26=>rem. 4 (Eureka!)
E.7p+6+32=>rem. 3
Ans=D
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Re: Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post 06 Aug 2018, 00:45
EgmatQuantExpert wrote:
e-GMAT Question of the Week #10


When an even integer p is divided by 7, the remainder is 6. Which of the following, when added to p, gives a sum that leaves a remainder of 4 when divided by 14?

Options

(a) 10
(b) 14
(c) 16
(d) 26
(e) 32

To access all the QOW questions: Question of the Week: Consolidated List
20( even num when divided by 7 gives remainder 6 ) + 26 = 46, divided be 14 leaves remainder 4.
Ans.D. 26

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Re: Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post 06 Aug 2018, 01:17
From the question -
P = 7I + 6 ......1)
P + x = 14I + 4 ......2)
Where I is a constant

Therefore, x=7I - 2 ...... From 1 & 2

From the options D) 7x4-2 = 26

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Re: Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post 06 Aug 2018, 07:29
1
[As Dividend = Divisor*Quotient + Remainder ]

P= 7q+6 ------> 1 [q is an integer]

Now ,Let S = 14t+4 -----> 2 [ t is an integer & S= P+ New number]

Equation 2- Equation 1

S-P = 14t+4-7q-6
P+New number-P = 7(q-2t)-2
New number= 7*interger-2
New number+2= 7*interger


Above equation implies that (new number + 2) is divisible by 7 , hence check which answer option +2 is divisible
by 7

ANS: D [ 26+2=28]
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Re: Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post 15 Aug 2018, 23:34

Solution


Given:
    • p is an even integer ………. (1)
    • When p is divided by 7, the remainder is 6 …………. (2)

To find:
    • Which of the given answer choices, when added to p, gives a sum that leaves a remainder of 4 when divided by 14

Approach and Working:
    • From statement (2), we can write ‘p’ in the form, N = Q*D + R, as:
      o p = 7a + 6,
    • And, from statement (1), we know that p is n even integer.
      o Implies, 7a + 6 is an even integer
      o Since, 6 is an even number, 7a must be even [even + even = even]
      o For 7a to be even,
         ‘a’ must be even, since, 7 is an odd number [odd * even = even]
      o Thus, let’s assume a = 2b, where b is a positive integer
      o Therefore, p = 7 * 2b + 6 = 14b + 6 ………………………………………. (3)
    • Now, let’s assume the number that has to be added to p as ‘n’,
      o Thus, the sum will be = p + n = 14b + 6 + n …………………………… (4)
      o This can be written as, 14b + (n+2) + 4
    • We are given that the sum leaves a remainder of 4, when divided by 14,
      o That is, 14b + (n+2) + 4 leaves a remainder of 4, when divided by 14
      o Implies, 14b + (n+2) must be divisible by 14
         14b is always divisible by 14
         Thus, n+2 must be divisible by 14
         For this, ‘n’ can take the values of the form, 12 + 14k, where k is an integer

Verifying the options, only option D, which is equal to 26, satisfies this condition.

Hence, the correct answer is option D.

Answer: D

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Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post 18 Aug 2018, 08:13
EgmatQuantExpert wrote:
e-GMAT Question of the Week #10


When an even integer p is divided by 7, the remainder is 6. Which of the following, when added to p, gives a sum that leaves a remainder of 4 when divided by 14?

Options

(a) 10
(b) 14
(c) 16
(d) 26
(e) 32


least value of p=6
let x=number added to p
q=quotient
6+x=14q+4→
14q-x=2
we know that x and q are even
if q=2, then x=26
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Re: Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post 18 Aug 2018, 09:21
EgmatQuantExpert wrote:
e-GMAT Question of the Week #10


When an even integer p is divided by 7, the remainder is 6. Which of the following, when added to p, gives a sum that leaves a remainder of 4 when divided by 14?

Options

(a) 10
(b) 14
(c) 16
(d) 26
(e) 32

To access all the questions: Question of the Week: Consolidated List

Image

Here is my approach.
Let x is number added.
p = 7a+6 , a is non-negative integer
p+x = 14b+4 , b is non-negative integer
--> 7a+6+x = 14b+4 --> x+2 = 14b-7a --> (x+2) is divisible by 7 --> Only D satisfies.
Answer D.
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Re: Question of the Week- 10 ( When an even integer p is divided by 7,...)  [#permalink]

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New post 18 Aug 2018, 10:29
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EgmatQuantExpert wrote:
e-GMAT Question of the Week #10


When an even integer p is divided by 7, the remainder is 6. Which of the following, when added to p, gives a sum that leaves a remainder of 4 when divided by 14?

Options

(a) 10
(b) 14
(c) 16
(d) 26
(e) 32


Least possible value of p = 6

NOw, check using options

(A) \(\frac{6 + 10}{14}\) = Remainder 2
(B) \(\frac{6 + 14}{14}\) = Remainder 6
(C) \(\frac{6 + 16}{14}\) = Remainder 8

(D) \(\frac{6 + 26}{14}\) = Remainder 4
(E) \(\frac{6 + 32}{14}\) = Remainder 8

Thus, Answer must be (D)
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Re: Question of the Week- 10 ( When an even integer p is divided by 7,...) &nbs [#permalink] 18 Aug 2018, 10:29
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