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# Help with Kaplan Number Properties Q

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Help with Kaplan Number Properties Q [#permalink]  08 Aug 2012, 19:47
Q How many three-digit positive integers can be divided by 2 to produce a new integer with the same tens digit and units digit as the original integer?
None
One
Two
Three
Four

The answer is four. Can you please explain in detail how this was achieved. Will really appreciate the help. Thank you!
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Re: Help with Kaplan Number Properties Q [#permalink]  09 Aug 2012, 03:33
1
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Expert's post
Q How many three-digit positive integers can be divided by 2 to produce a new integer with the same tens digit and units digit as the original integer?
None
One
Two
Three
Four

The answer is four. Can you please explain in detail how this was achieved. Will really appreciate the help. Thank you!

When will the digits stay the same even after division by 2? The original number has to be divisible by 2 since the new number is also an integer.
Think: 242 divided by 2 gives 121. 546 gives 273 etc
The last two digits will not be the same. There must be some special property of last 2 digits for them to stay the same. Also, there are very few such numbers since answer is 0/1/2/3/4.

If the last two digits are 0s, they might remain 0s.
200 comes to mind. 200 divided by 2 gives 100. Last two digits are same.
What about 300? 300/2 = 150
So the hundreds digit must be even. Therefore, the only numbers are 200, 400, 600, 800.
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Intern
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Posts: 4
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Re: Help with Kaplan Number Properties Q [#permalink]  09 Aug 2012, 15:46
VeritasPrepKarishma wrote:
Q How many three-digit positive integers can be divided by 2 to produce a new integer with the same tens digit and units digit as the original integer?
None
One
Two
Three
Four

The answer is four. Can you please explain in detail how this was achieved. Will really appreciate the help. Thank you!

When will the digits stay the same even after division by 2? The original number has to be divisible by 2 since the new number is also an integer.
Think: 242 divided by 2 gives 121. 546 gives 273 etc
The last two digits will not be the same. There must be some special property of last 2 digits for them to stay the same. Also, there are very few such numbers since answer is 0/1/2/3/4.

If the last two digits are 0s, they might remain 0s.
200 comes to mind. 200 divided by 2 gives 100. Last two digits are same.
What about 300? 300/2 = 150
So the hundreds digit must be even. Therefore, the only numbers are 200, 400, 600, 800.

Thank you for your help, Karishma. I really appreciate it.
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Re: Help with Kaplan Number Properties Q [#permalink]  17 Aug 2012, 10:33
Q How many three-digit positive integers can be divided by 2 to produce a new integer with the same tens digit and units digit as the original integer?
None
One
Two
Three
Four

The answer is four. Can you please explain in detail how this was achieved. Will really appreciate the help. Thank you!

We are looking for a three digit number $$abc$$ such that $$abc = 2*xbc$$, where $$xbc$$ is also a three digit number.
This means $$100a+10b+c=200x+20b+2c$$, from which $$100(a-2x)=10b+c.$$
Since $$0\leq10b+c\leq99$$ and $$100(a-2x)\geq0$$, the only possibility is $$b=c=0$$ and $$a=2x.$$
So, $$a$$ can 2, 4, 6, or 8.

Answer E (total of 4 numbers - $$200, 400, 600, 800$$).
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Re: Help with Kaplan Number Properties Q   [#permalink] 17 Aug 2012, 10:33
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