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How many integers k greater than 100 and less than 1000 are [#permalink]
17 Sep 2010, 04:39

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Question Stats:

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How many integers k greater than 100 and less than 1000 are there such that if the hundreds and the unit digits of k are reversed, the resulting integer is k + 99?

How many integers k greater than 100 and less than 1000 are there such that if the hundreds and the unit digits of k are reverced, the resulting integer is k + 99?

50 60 70 80 90

please provide explaination .. with minimal calculations

Some wording. Anyway:

\(k\) is 3-digit integer. Any 3 digit integer can be expressed as \(100a+10b+c\).

Now, we are told that \((100a+10b+c)+99=100c+10b+a\) --> \(99c-99a=99\) --> \(c-a=1\) --> there are 8 pairs of \(c\) and \(a\) possible such that their difference to be 1: (9,8), (8,7), ..., and (2,1) (note that \(a\) can not be 0 as \(k\) is 3-digit integer and its hundreds digit, \(a\), is more than or equal to 1).

Also, \(b\), tens digit of \(k\), could tale 10 values (0, 1, ..., 9) so there are total of 8*10=80 such numbers possible.

How many integers k greater than 100 and less than 1000 are there such that if the hundreds and the unit digits of k are reverced, the resulting integer is k + 99?

50 60 70 80 90

please provide explaination .. with minimal calculations

Not sure if this is the shortest.. But this is how I did this

There are 8 sets of integers with hundreds and units digits exchanged that satisfies k + 99. 1. 102 | 201 (satisfies k+99, where k = 102) 2. 203 | 302 (satisfies k+99, where k = 203) 3. ... 4. ... 5. ... 6. ... 7. 708 | 807 8. 809 | 908

Re: How many integers k greater than 100 and less than 1000 are [#permalink]
10 Jan 2014, 07:46

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How many integers K greater that 100 and less than 1000 are [#permalink]
15 May 2014, 12:45

How many integers K greater that 100 and less than 1000 are there such that if the hundreds and the units digits of K are reversed, the resulting integer is K+99?

Re: How many integers K greater that 100 and less than 1000 are [#permalink]
15 May 2014, 17:22

2

This post received KUDOS

Ans is D. The hundredth digit is always smaller from units digit by 1 as obtained by solving the equation 100a+10b+c+99=100c+10b+a. So 'a' can have values from 1 to 8 and for every value of a, 'b' varies from 0 to 9. So total integers are 8×10.

Re: How many integers K greater that 100 and less than 1000 are [#permalink]
15 May 2014, 17:22

tinku21rahu wrote:

How many integers K greater that 100 and less than 1000 are there such that if the hundreds and the units digits of K are reversed, the resulting integer is K+99?

a) 50 b) 60 c) 70 d) 80 e) 90

I think the answer is D - 80 integers. My logic is as follows:

Since we are adding 99, the hundreds digit will need to be less than the ones digit if the corresponding number when the two digits flipped is to be larger. Since the change in magnitude is just shy of 100, the different between the two can only be one.

Look at an example case of 100 < X < 200

102 -> 201

112 -> 211

...

192 -> 291

What about for 800 < X < 900?

809 -> 908

819 -> 918

...

899 -> 998

For each of these samples, there are 10 numbers between the two stated bounds (the tens digit ranges from 0 to 9). Since we can't do this for 900 < X < 1,000, there are 8 sets of 10, or 80 total integers.

Hopefully that makes sense (and correct!). Let me know if I'm missing something here!

Re: How many integers K greater that 100 and less than 1000 are [#permalink]
16 May 2014, 07:15

GordonFreeman: What you said said is 100% correct and that was the way i solved it, but that took me lot of time, Do you feel there is any easier was like PuneetSood Said.

PuneetSood: What you said seems to be an Easy Way. Thank You!

Re: How many integers K greater that 100 and less than 1000 are [#permalink]
16 May 2014, 07:43

tinku21rahu wrote:

GordonFreeman: What you said said is 100% correct and that was the way i solved it, but that took me lot of time, Do you feel there is any easier was like PuneetSood Said.

PuneetSood: What you said seems to be an Easy Way. Thank You!

PuneetSood's method is definitely slicker than mine. I only started my GMAT journey this week so when I approach problems like this I like to see all the moving parts. After I am more comfortable with a problem type I would opt for an approach like PuneetSood's.

Re: How many integers K greater that 100 and less than 1000 are [#permalink]
16 May 2014, 18:13

tinku21rahu wrote:

GordonFreeman: What you said said is 100% correct and that was the way i solved it, but that took me lot of time, Do you feel there is any easier was like PuneetSood Said.

PuneetSood: What you said seems to be an Easy Way. Thank You!

From here, we can assume that the hundreds digit and units digit must be 1 digit apart in order to satisfy this condition.

Only 9 options are possible: 9x8,8x7,7x6,6x5,5x4,4x3,3x2,2x1,1x0

We are also told in the question stem that we are restricted among 3 digits only which eliminates the 1x0 option. Now, there are only 8 possible options left. Since there is no restriction in the tens digits,

Re: How many integers K greater that 100 and less than 1000 are [#permalink]
26 May 2014, 23:59

Expert's post

tinku21rahu wrote:

How many integers K greater that 100 and less than 1000 are there such that if the hundreds and the units digits of K are reversed, the resulting integer is K+99?

a) 50 b) 60 c) 70 d) 80 e) 90

Merging similar topics. Please refer to the discussion above. _________________

Re: How many integers k greater than 100 and less than 1000 are [#permalink]
27 May 2014, 16:04

Let the 3 digit number be xyz If the units and hundred's digit is reversed then the number becomes zyx

Given:- 100z + 10y + x = 100x + 10y + z + 99

100z + 10y + x = 100x + 10y + z + 99

or, 99z - 99x = 99 or, 99(z-x)=99 or, z=x+1

which means z can take only 1 value i.e 1 more than x

Now in the original number xyz, y can take 10 values 0 through 9 x can take 1-8(8 values) and for each x, z can take exactly 1 value (from 2 through 9)

so 8 (values for x and z) X 10 (values of y) = 80. Hence D

gmatclubot

Re: How many integers k greater than 100 and less than 1000 are
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27 May 2014, 16:04

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