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The remainder when N is divided by 18 is 16. Given that N is

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The remainder when N is divided by 18 is 16. Given that N is [#permalink] New post 26 Apr 2013, 02:28
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The remainder when N is divided by 18 is 16. Given that N is a multiple of 28, how many integers between 0 and 18 inclusive could be the remainder when N/4 is divided by 18?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Just nice problem from http://www.mualphatheta.org/National_Co ... Tests.aspx
I know some ways how to solve it quickly. May be someone knows nicer way of solution. Thanks:)
[Reveal] Spoiler: OA

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Re: The remainder when N is divided by 18 is 16 [#permalink] New post 26 Apr 2013, 03:18
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The remainder when N is divided by 18 is 16, translated : N=18k+16
\frac{N}{4} is divided by 18 means what is the remainder of \frac{N}{4*18}?
Given that N is a multiple of 28, translated: N=28m

\frac{N}{4*18} with N=28m is \frac{28m}{4*18} or \frac{7m}{18} and its "form" can be written as 7m=18q+R ( or 14m=36q+2R, this will be useful later)

Going back to the first equation N=18k+16 = 28m=18k+16 = 14m=9k+8. From the equation before is its "useful" form 14m=36q+2R
so puttin them together 9k+8=36q+2R all the numbers k,q,R must be integer

8-2R=36q-9k
if q and r are 0 8-2R=0 so R=4 value #1
the other possible value of R (because must be positive, it's a reminder) will be in the case 9k>36q
The difference 36q-9k can be (36-45) = -9 but 8-2R=-9 means R=17/2 no integer
difference -18 => R = 5 value #2
difference -27 => R = 33/2 no integer
difference -36 => R=21 out of range 0,18
We can stop here bigger differences mean R out of 0,18 range

2 values, B
(I am not sure of my method though, the Master Mind could help here and +1 to the question! it took me 10 minutes to came up with a solution!)
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Re: The remainder when N is divided by 18 is 16 [#permalink] New post 26 Apr 2013, 03:46
Zarrolou wrote:
The remainder when N is divided by 18 is 16, translated : N=18k+16
\frac{N}{4} is divided by 18 means what is the remainder of \frac{N}{4*18}?
Given that N is a multiple of 28, translated: N=28m

\frac{N}{4*18} with N=28m is \frac{28m}{4*18} or \frac{7m}{18} and its "form" can be written as 7m=18q+R ( or 14m=36q+2R, this will be useful later)

Going back to the first equation N=18k+16 = 28m=18k+16 = 14m=9k+8. From the equation before is its "useful" form 14m=36q+2R
so puttin them together 9k+8=36q+2R all the numbers k,q,R must be integer

8-2R=36q-9k
if q and r are 0 8-2R=0 so R=4 value #1
the other possible value of R (because must be positive, it's a reminder) will be in the case 9k>36q
The difference 36q-9k can be (36-45) = -9 but 8-2R=-9 means R=17/2 no integer
difference -18 => R = 5 value #2
difference -27 => R = 33/2 no integer
difference -36 => R=21 out of range 0,18
We can stop here bigger differences mean R out of 0,18 range

2 values, B
(I am not sure of my method though, the Master Mind could help here and +1 to the question! it took me 10 minutes to came up with a solution!)


Thank you so much for solution and kudos!

It took me some time to find the nice solution. I will post how I see the solution later here. I'm just waiting for possible other comments.
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Re: The remainder when N is divided by 18 is 16 [#permalink] New post 26 Apr 2013, 04:49
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The remainder when N is divided by 18 is 16. Given that N is a multiple of 28, how many integers between 0 and 18 inclusive could be the remainder when
\frac{N}{4} is divided by 18?

Let N = 28x
so 28x = 18y + 16 or 18z - 2 both are equivalent .
so 28x = 18z -2 according to statement mentioned .

Now remainder when N/4 is divided by 18
let remainder be R
Let N/4 = 18q + R
Substituting N = 28x = 18z-2 we get
18z -2 = 72q + 4R
therefore R = (18(z - 4q)-2)/4 = (9(z - 4q ) - 2 ) /2 = (9*someinteger - 1) /2
If a number is divided by 18 so remainder is between 1 and 17 .
Substituting integer values we get :
(9*1 -1)/2 = 4 possible remainder
(9*2 -1 )/2 = 8.5 not possible
(9*3 -1 )/2 = 13 possible
(9*4 -1 )/2 = 17.5 not possible

Thus we get only 2 possible values for remainder i.e 4 and 13 hence answer is 2 .
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Re: The remainder when N is divided by 18 is 16. Given that N is [#permalink] New post 27 Apr 2013, 01:10
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So, my solution. Just a little bit different from the previous.


The remainder when N is divided by 18 is 16 means that N=18q+16 for some integer q.
N is a multiple of 28 means that N=28s for some integer s.

We need to find the remainder when \frac{N}{4} is divided by 18.

On one hand \frac{N}{4}=7s, on the other hand \frac{N}{4}=\frac{9q}{2}+4. Since 7s=\frac{9q}{2}+4 and s is an integer, q must be even.

So, \frac{N}{4}=9k+4 for some integer k.
Ifk is even (k=2n for some integer n) the remainder when \frac{N}{4} is divided by 18 is 4 (\frac{N}{4}=9*2n+4=18n+4).
If k is odd (k=2n+1 for some integer n) the remainder when \frac{N}{4} is divided by 18 is 13 (\frac{N}{4}=9(2n+1)+4=18n+13).

So, there two possible values for the remainder 4 and 13.
The answer is B.
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Re: The remainder when N is divided by 18 is 16. Given that N is   [#permalink] 27 Apr 2013, 01:10
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The remainder when N is divided by 18 is 16. Given that N is

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