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How many three-digit integers can be divided by 2 to produce

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How many three-digit integers can be divided by 2 to produce [#permalink] New post 22 Sep 2005, 12:01
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How many three-digit integers can be divided by 2 to produce a new integer with the same tens digit and units digit as the original integer?

A. None
B. One
C. Two
D. Three
E. Four
[Reveal] Spoiler: OA

Last edited by Bunuel on 21 Feb 2012, 00:00, edited 1 time in total.
Edited the question and added the OA
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Re: How many three-digit integers can be divided by 2 to produce [#permalink] New post 21 Feb 2012, 00:11
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acid_burn wrote:
How many three-digit integers can be divided by 2 to produce a new integer with the same tens digit and units digit as the original integer?

A. None
B. One
C. Two
D. Three
E. Four


The tens and units digits of a three-digit integer must be 0: X00 (no digit gives the same digit when divided by 2 but 0: 0/2=0). Now, apart from this, X must be even because 300/2=150, not the same tens and units digits. Hence there are following three-digit integers possible: 200, 400, 600, and 800.

Answer: E.
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Re: How many three-digit integers can be divided by 2 to produce [#permalink] New post 02 Jul 2013, 00:15
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Re: How many three-digit integers can be divided by 2 to produce [#permalink] New post 02 Jul 2013, 16:05
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acid_burn wrote:
How many three-digit integers can be divided by 2 to produce a new integer with the same tens digit and units digit as the original integer?

A. None
B. One
C. Two
D. Three
E. Four


hi,
3 digit number divided by 2==>this will yield either a 2 digit number or a 3 digit number

case1:if it yields a 2 digit number
let three digit number =xyz
therefore 2 digit no.=yz===>as unit and tens digit are same
now
we can write :
100x+10y+z=2*(10y+z)
solving
100x = 10y+z
clearly this is not possible as LHS is a 3 digit number and RHS is a 2 digit number.

case 2: if it yields a 3 digit number lets say ayz
then we can write
100x+10y+z=2(100a+10y+z)
solving
100(x-2a)=10y+z
this is only possible if 10y+z =0...and x-2a=0
it means y=z=0..and x=2a
therefore possible values are
put a=1=>x=2=>number=200
a=2=>x=4=>number=400
a=3=>x=6=>number=600
a=4=>x=8=>number=800
a=5=>x=10=>number=1000(its a 4 digit hence not possible)
so only 4 numbers are possible.
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Re: How many three-digit integers can be divided by 2 to produce [#permalink] New post 02 Jul 2013, 18:41
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Two conditions have to be met for the units digit of a 2 digit number to be the same when divided by 2:

1) The units digit of the original number must be 0.
2) The tens digit of the original number has to be even.

The same conditions can be extended to a 3 digit number if both the units digit and the tens digit of the new number have to be the same.

That is,

1) the units digit and tens digit of the original number have to be 0.
2) the hundreds digit of the original number has to be even

So these numbers are 200, 400, 600 and 800

Thus 4 numbers are possible.
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Re: How many three-digit integers can be divided by 2 to produce [#permalink] New post 24 Aug 2014, 22:34
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Re: How many three-digit integers can be divided by 2 to produce   [#permalink] 24 Aug 2014, 22:34
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