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number prop

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number prop [#permalink] New post 30 Jun 2010, 06:01
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z divided by 12 leaves remainder of 9. What will be remainder when a z divided by 24.
How do we approach such problems?
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Re: number prop [#permalink] New post 30 Jun 2010, 06:16
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I'm not really sure if this is entirely correct, but here's my take on this question.

It's given that z leaves a reminder of 9 when divided by 12. This means:

z=12x+9

So now dividing z by 24 we get:

\frac{z}{24} = \frac{12x}{24} + 9{24}

Any number can be split into the sum of two numbers and reminder can be analyzed individually, just as long as the reminder value is not more than the divisor.

That is, if we want to find the reminder when 45 is divided by 12, we can do 45 = 36 + 9 and say, 36/12 = 3, so no reminder, and 9/12 will leave reminder 9 and hence 45/12 leaves reminder 9.

Similarly here, we have \frac{12x}{24} which can be reduced to \frac{x}{2}. x can either be an odd number or even number and subsequently leave a reminder of 1 or 0.

Then we have \frac{9}{24} which will leave a reminder of 9.

So the net reminder when z is divided by 24 is either a 9 or 10.

What's the OA?
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Re: number prop [#permalink] New post 30 Jun 2010, 06:20
Well I don't have the OA for this. I was asked by one of the colleague of mine who is preparing for gmat and I was stumped!
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Re: number prop [#permalink] New post 30 Jun 2010, 06:20
Ah, if you can please ask him/her for the answer and post it. I'd like to know if my approach was right. :)
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Re: number prop [#permalink] New post 30 Jun 2010, 06:28
A quick check with him and it seems he just knows the question, but no answer.
Guess we will have to wait for some number prop wizard to verify ..
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Re: number prop [#permalink] New post 30 Jun 2010, 06:45
I think the answer cant be determined with the given info.

Lets z=45, then if its divided by 12, remainder is 9, if its divided by 24, remainder is 21.
Now lets z= 33, divided by 12 & 24 gives the same remainder 9.

Hence we cant determine the answer from the given info.
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Re: number prop [#permalink] New post 30 Jun 2010, 07:52
Let's say the two remainders are r and R, so we have that

z= 12*x + r
z= 24*y + R
where 0<r<11 and 0<R<23.

Subtracting one equation from the another we obtain

0= 12*(x-2*y) + r - R => R= 12*(x-2*y) + r

Now it must be 0<=(x-2*y)<2 in order to be 0<R<23 = > R=r or R=12+r

In our case r=9 so It could be R=9 or R=21 .. but only one of these two choices.

@whiplash2411

You are right, remainders are good guys: if you want to find the remainder of a sum you can calculate the sum of the remainders ( ... and eventually calculate again the remainder if necessary ) BUT you cannot mix the numbers you are dividing by.
Quote:
Similarly here, we have \frac{12x}{24} which can be reduced to \frac{x}{2} . x can either be an odd number or even number and subsequently leave a reminder of 1 or 0.

Then we have \frac{9}{24} which will leave a reminder of 9.

You cannot consider the sum of the remainder of \frac{9}{24} and the remainder of \frac{x}{2} ... it will be the remainder of z divided by what... 24 or 2 ? :)
So following your approach if x is even then the remainder of \frac{12x}{24} will be 0, if x is odd the remainder will be 12 ( not 1).
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Re: number prop [#permalink] New post 30 Jun 2010, 08:49
Hey udini,

That's what I meant, sorry :D (Got lost in translation)
Re: number prop   [#permalink] 30 Jun 2010, 08:49
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