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 Post subject: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:45 am 
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Given that R is a positive three-digit integer, what is the hundreds digit of R?
(1) The hundreds digit of 3R is 8.
(2) (R + 1) results in a number with the hundreds digit of 9.
A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(C) Statement (1) tells us that if we multiply the number R by 3, the hundreds digit will be 8. If R were 270, 3R would be 810, and if R were 600, 3R would be 1800. In both cases, Statement (1) is true, but the hundreds digit of R is not the same in these two cases. So this statement is insufficient.

Statement (2) tells us that if we add 1 to R, then the hundreds digit will be 9. So R could be any integer from 899 to 998, since adding 1 to any of these numbers will yield a number with 9 as the hundreds digit. So the hundreds digit of R could be 8 or 9, and Statement (2) is insufficient.

Together the statements are sufficient. Statement (2) tells us that R must be between 899 and 998, and Statement (1) tells us that 3R has a hundreds digit of 8. The only integers in the given range that can be multiplied by three to yield an integer with a hundreds digit of 8 are the numbers from 934 to 966, because:
3 × 934 = 2802 and 3 × 966 = 2898.
So the hundreds digit is 9.

Since both statements are insufficient individually, but sufficient when combined, the correct answer is choice (C).
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This is a hard question, how can you come up with 270 as the first number to start with? For such type of question, is there any hint or tips of how to tackle? Thanks.


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 Post subject: Re: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:46 am 
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This question is a hard one indeed, but it is NOT as hard, as it seems.

When dealing with such type of questions it's good to keep in mind the range of possible values:

1. The base statement tells us "R is a positive three-digit integer", so the initial range is 100, ..., 999. Obviously, there're 9 options of the hundreds digit.

2. The second statement is very easy to analyze. It tells us "(R + 1) results in a number with the hundreds digit of 9", so (R + 1) can be any integer from 900 to 999 inclusive. Therefore the range of possible values of R is from 899 to 998. Obviously there're 2 options for the hundreds digit: 8 and 9. Also note, that option 8 is represented by one number only.

3. The first statement tells us "The hundreds digit of 3R is 8". The range for R in this case is not that obvious, but we should start with the question "what is the range of possible values for 3R based on the base statement (initial range for R)?". Since the range for R is [100, 999], then the range for 3R [300, 2997]. (You do NOT need to calculate 2997, keeping in mind "couple units less than 3000" is enough). Clearly, in this range there are numbers [800, 899], [1800, 1899] and even [2800, 2899] that apparently correspond to values of R that significantly differ, by hundreds, and so they have different possible values of the hundreds digit. The numbers 270 and 600 in the solution are just two possible ones, which picked from a wide ranges: integers from (800/3, 899/3), [1800/3, 1899/3], (2800/3, 2899/3).

4. When we combine the first and the second statements, we just need to exclude he number 899 from the range given by the second statement. We can easily exclude it as 899 × 3 = 900 × 3 - 3 = 2700 - 3 = 2697. Therefore the only possible option is 9. The both statements are sufficient when combined.

To summarize, the reasoning is NOT that hard as it seems at first. We deal with ranges that are very simple for the base statement and for the second statement. Reasoning the first statement is harder, but starting it with the simple range for 3R it's not that long to find the possible ranges for R itself. Combining the statements, we easily exclude 899 from the range set by the second statement and get the final answer.


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 Post subject: Re: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:46 am 
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For statement 2, you mention that if you add 1 to R, R could be any integer from 899 to 998. What if R is 1899? When you add 1, the hundreds digit is also 9. Would this make the answer E?


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 Post subject: Re: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:48 am 
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questioner wrote:
What if R is 1899?
It can NOT be 1899, because the questions statement sets it up as a three-digit number:
Quote:
... R is a positive three-digit integer ...


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 Post subject: Re: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:49 am 
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I'm a little confused as I thought the question states that the number is a 3 digit not 4 digit integer? And therefore can only start with a 2, so the answer is A.
Many thanks.


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 Post subject: Re: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:50 am 
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questioner wrote:
I'm a little confused as I thought the question states that the number is a 3 digit not 4 digit integer?
Number R is a 3-digit integer, but 3R is not limited to 3 digits. 3R can be a 4-digit integer.


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 Post subject: Re: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:50 am 
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The solution seems wrong since the question stem says that it is a 3 digit number. The right solution should be A because there is only one possibility.

As far as B goes that is insufficient because it could be 899 as well as 900.


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 Post subject: Re: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:51 am 
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Quote:
As far as B goes that is insufficient because it could be 899 as well as 900.
That is correct.

Quote:
The solution seems wrong since the question stem says that its a 3 digit number.
Number R is a 3-digit integer, but 3R is not limited to 3 digits. 3R can be a 4-digit integer.


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 Post subject: Re: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:52 am 
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Sorry, I am still confused. How can it be that you say that the hundreds digit of 810 is 8, but that it is not the same as in 1800. As i understand it, in 1800, the hundreds digit is still 8. Isn't it?


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 Post subject: Re: GMAT Number Theory
PostPosted: Fri May 03, 2013 8:54 am 
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rquerub wrote:
Sorry, I am still confused. How can it be that you say that the hundreds digit of 810 is 8, but that it is not the same as in 1800. As i understand it, in 1800, the hundreds digit is still 8. Isn't it?
1800 and 810 have the same hundreds digit, 8. But these are the possibilities for 3R, not R itself. The possibilities for R are 600 and 270. These numbers have different hundreds digits, 6 and 2 respectively.


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