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 Post subject: GMAT Number Theory (Data Sufficiency)
PostPosted: Sun Dec 12, 2010 4:57 am 
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Given that R is a positive integer, what is the hundreds digit of R?
(1) The hundreds digit of 3R is 8.
(2) The hundreds digit of (R + 1) is 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 is 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 we can't determine the hundreds digit of R. 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 number whose last three digits range 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.

Combining the two statements helps us to narrow down the hundreds digit of R even further. Statement (2) tells us that the last three digits of R must be between 899 and 998. The only numbers in this range that can be multiplied by three and yield a number that has a hundreds digit of 8 are the numbers 934 to 966 (3 × 934 = 2802 and 3 × 966 = 2898).
So the statement combined are sufficient, because we now know that the hundreds digit of R will be 9. The answer is (C).
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Hi,
In your solution you have used 3 digit number for R. But the question does not give any information about how many digits are in R. R can be greater than equal to 3 digits.

Could you please let me know your thoughts?


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 Post subject: Re: math (test 2, question 24): number theory, data sufficie
PostPosted: Sun Dec 12, 2010 5:15 am 
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Joined: Sun May 30, 2010 2:23 am
Posts: 498
We do NOT assume in explanation that R is a 3-digit integer. R can have any number of digits, but we analyse only the last three digits.

We do so, because the hundreds digit of the resulting number in case of multiplication by 3 or addition of 1 depends solely on the last three digits of the original number.

Suppose R = ab....cxyz, where each letter represents a digit, and there could be any number of any digits in between b and c. We can rewrite R as:
R = ab....cxyz = ab....c000 + xyz

We can see that digits ab....c do not affect hundreds digit if R is multiplied by an an integer or if to add a positive number.


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