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Positive integers x and y leave remainders 5 and 8 respectively when

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Positive integers x and y leave remainders 5 and 8 respectively when  [#permalink]

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New post 24 Oct 2017, 01:05
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Positive integers x and y leave remainders 5 and 8 respectively when divided by 11. Which of the following is not a possible value of x + 2y?

A. 120
B. 196
C. 219
D. 351
E. 560

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Re: Positive integers x and y leave remainders 5 and 8 respectively when  [#permalink]

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New post 24 Oct 2017, 01:15
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nkmungila wrote:
Positive integers x and y leave remainders 5 and 8 respectively when divided by 11. Which of the following is not a possible value of x + 2y?

A. 120
B. 196
C. 219
D. 351
E. 560


It should be B

We can rewrite both x and y as

\(x = 11m + 5\)
\(y = 11p + 8\)

where m and n are non negative integers.

\(x + 2y = 11m + 5 + 2 (11p +8)\)
\(x + 2y = 11m + 22p + 5 + 16\)
\(x + 2y = 11m + 22p + 21\)

We can now rewrite it as

\(x + 2y = 11m + 22p + 11 + 10\)

First 3 terms are multiples of 11 and an additional 10. So this expression results in a number that is 10 more than a multiple of 11. In other words, dividing the number would result in a remainder of 10. Now looking at options.

A. \(120 = 11*10 + 10\)
B. \(196 = 11*17 + 9\)
C. \(219 = 11*19 + 10\)
D. \(351 = 11*31 + 10\)
E. \(560 = 11*50 + 10\)
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Re: Positive integers x and y leave remainders 5 and 8 respectively when  [#permalink]

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New post 13 Jan 2019, 10:46
x= 11a+5
y= 11b+8
resultant x+2y = 11a+5+22b+16 = 11(a+2b)+21
or 11(a+b+1) +10
thus the resultant should be or form leaving remainder 10.
Evaluating all options only B does not fulfill the criteria
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Re: Positive integers x and y leave remainders 5 and 8 respectively when  [#permalink]

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New post 13 Jan 2019, 11:13
nkmungila wrote:
Positive integers x and y leave remainders 5 and 8 respectively when divided by 11. Which of the following is not a possible value of x + 2y?

A. 120
B. 196
C. 219
D. 351
E. 560


When you try to get the values for x and y by substituting values for n, you will notice they 3 apart from one another.
n=0,1,2,3,4
---------------0------1-----2-----3
x= 11n + 5--5-----16----27----38 and so on
y= 11n + 8--8-----19----30----41 and so on

You can start from center option C, or eliminate one by one, i did the latter.
x=60 y=63
x+2y = 186

Answer B
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Positive integers x and y leave remainders 5 and 8 respectively when  [#permalink]

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New post 13 Jan 2019, 17:13
nkmungila wrote:
Positive integers x and y leave remainders 5 and 8 respectively when divided by 11. Which of the following is not a possible value of x + 2y?

A. 120
B. 196
C. 219
D. 351
E. 560


x+y=11q+5+11p+8→
x+2y-21=11(q+2p)
so x+2y-21 must be a multiple of 11
subtracting 21 from each answer choice,
the only non-multiple of 11 is 196-21=175
B
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Positive integers x and y leave remainders 5 and 8 respectively when   [#permalink] 13 Jan 2019, 17:13
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