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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]

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26 Apr 2013, 03:28
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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]

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26 Apr 2013, 04:18
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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]

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26 Apr 2013, 04:46
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]

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26 Apr 2013, 05:49
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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]

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27 Apr 2013, 02:10
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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 $$k$$.
If$$k$$ 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.
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Re: The remainder when N is divided by 18 is 16. Given that N is [#permalink]

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29 Jul 2014, 01:30
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Re: The remainder when N is divided by 18 is 16. Given that N is [#permalink]

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11 Oct 2015, 06:01
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Re: The remainder when N is divided by 18 is 16. Given that N is   [#permalink] 11 Oct 2015, 06:01
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