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The three-digit positive integer n can be written as ABC

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The three-digit positive integer n can be written as ABC [#permalink] New post 03 Jul 2010, 01:11
00:00
A
B
C
D
E

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(N/A)

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33% (01:26) correct 67% (01:57) wrong based on 21 sessions
not sure if anyone is posting this questions from the mbamission.com quest....

The three-digit positive integer n can be written as ABC, in which A, B, and C stand for the unknown digits of n. What is the remainder when n is divided by 37?

(1) A + B/10 + C/100 = B + C/10 + A/100

(2) A + B/10 + C/100 = C + A/10 + B/100

OA:
[Reveal] Spoiler:
D
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Re: The Quest for 700: Weekly GMAT Challenge [#permalink] New post 03 Jul 2010, 01:12
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OA answer
[Reveal] Spoiler:
The question stem tells us that the positive integer n has three unknown digits: A, B, and C, in that order. In other words, n can be written as ABC. Note that in this context, ABC does not represent the product of the variables A, B, and C, but rather a three-digit integer with unknown digit values. It is important to note that since A, B, and C stand for digits, their values are restricted to the ten digits 0 through 9. Moreover, A cannot equal 0, since we know that n is a “three-digit” integer and therefore must be at least 100.

We are asked for the remainder after n is divided by 37. We could rephrase this question in a variety of ways, but none of them are particularly better than simply leaving the question as is.

Statement (1): SUFFICIENT. We can translate this statement to a decimal representation, which will be easier to understand. The left side of the equation, in words, is “A units plus B tenths plus C hundredths.” We can write this in shorthand: A.BC (that is, “A point BC”). After performing the same translation to the right side of the equation, we can see that we get the following:

A.BC = B.CA

Since A, B, and C stand for digits, we can match up the decimal representations and observe that A = B and B = C. Thus, all the digits are the same.

This means that we can write n as AAA, which is simply 111 × A.

Now, 111 factors into 3 × 37, so n = 3 × 37 × A. Thus, n is a multiple of 37, and the remainder after division by 37 is zero.

Statement (2): SUFFICIENT. Again, we can match up the decimal representations of the given equation and find that all the digits are the same. The logic from that point forward is identical to that shown above.

The correct answer is (D): EACH statement ALONE is sufficient.
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Re: The Quest for 700: Weekly GMAT Challenge [#permalink] New post 03 Jul 2010, 01:16
my answer method
[Reveal] Spoiler:
(1) is simplified to 100A +10B+C = 100B+10C+A
therefore 99A=90B+C
11A=10B+C

thus 10 b + c gives 11 a ----> a,b,c similar numbers

if a,b,c are 111 therefore 37 has remainder 0....if 222, then remainder 0 again coz 222 is multiple of 111
same for 333,444 etc

(2) end up with similar equation thus similar answer

OA: D
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Re: The Quest for 700: Weekly GMAT Challenge [#permalink] New post 03 Jul 2010, 06:23
Both statements indicate that the three digits are the same. Factor 111 into 37 and 3 - remainder is 0 - any other numbers with all the same digits are multiples of 37. Therefore D. Intimidating phrasing but a fun problem in the end.
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Re: The Quest for 700: Weekly GMAT Challenge [#permalink] New post 03 Jul 2010, 09:09
Great question! I chose E initially as there were multiple numbers (111,222,333 etc) for which condition 1 and two hold good. However when u divide each of these by 37, you realize that the remainder is always Zero. Hence D is right.
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Re: The Quest for 700: Weekly GMAT Challenge [#permalink] New post 09 Jul 2010, 20:54
zisis wrote:
OA answer
[Reveal] Spoiler:
The question stem tells us that the positive integer n has three unknown digits: A, B, and C, in that order. In other words, n can be written as ABC. Note that in this context, ABC does not represent the product of the variables A, B, and C, but rather a three-digit integer with unknown digit values. It is important to note that since A, B, and C stand for digits, their values are restricted to the ten digits 0 through 9. Moreover, A cannot equal 0, since we know that n is a “three-digit” integer and therefore must be at least 100.

We are asked for the remainder after n is divided by 37. We could rephrase this question in a variety of ways, but none of them are particularly better than simply leaving the question as is.

Statement (1): SUFFICIENT. We can translate this statement to a decimal representation, which will be easier to understand. The left side of the equation, in words, is “A units plus B tenths plus C hundredths.” We can write this in shorthand: A.BC (that is, “A point BC”). After performing the same translation to the right side of the equation, we can see that we get the following:

A.BC = B.CA

Since A, B, and C stand for digits, we can match up the decimal representations and observe that A = B and B = C. Thus, all the digits are the same.

This means that we can write n as AAA, which is simply 111 × A.

Now, 111 factors into 3 × 37, so n = 3 × 37 × A. Thus, n is a multiple of 37, and the remainder after division by 37 is zero.

Statement (2): SUFFICIENT. Again, we can match up the decimal representations of the given equation and find that all the digits are the same. The logic from that point forward is identical to that shown above.

The correct answer is (D): EACH statement ALONE is sufficient.


zisis, great explanation. Kudos to you.
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Re: The Quest for 700: Weekly GMAT Challenge [#permalink] New post 10 Jul 2010, 11:51
zisis wrote:
The question stem tells us that the positive integer n has three unknown digits: A, B, and C, in that order. In other words, n can be written as ABC. Note that in this context, ABC does not represent the product of the variables A, B, and C, but rather a three-digit integer with unknown digit values. It is important to note that since A, B, and C stand for digits, their values are restricted to the ten digits 0 through 9. Moreover, A cannot equal 0, since we know that n is a “three-digit” integer and therefore must be at least 100.

We are asked for the remainder after n is divided by 37. We could rephrase this question in a variety of ways, but none of them are particularly better than simply leaving the question as is.

Statement (1): SUFFICIENT. We can translate this statement to a decimal representation, which will be easier to understand. The left side of the equation, in words, is “A units plus B tenths plus C hundredths.” We can write this in shorthand: A.BC (that is, “A point BC”). After performing the same translation to the right side of the equation, we can see that we get the following:

A.BC = B.CA

Since A, B, and C stand for digits, we can match up the decimal representations and observe that A = B and B = C. Thus, all the digits are the same.

This means that we can write n as AAA, which is simply 111 × A.

Now, 111 factors into 3 × 37, so n = 3 × 37 × A. Thus, n is a multiple of 37, and the remainder after division by 37 is zero.

Statement (2): SUFFICIENT. Again, we can match up the decimal representations of the given equation and find that all the digits are the same. The logic from that point forward is identical to that shown above.

The correct answer is (D): EACH statement ALONE is sufficient.


+1
It took me more then 3 minutes to solve this...
Nice explanation..
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Re: The Quest for 700: Weekly GMAT Challenge [#permalink] New post 27 Jul 2010, 03:16
Interesting question, it help me recalling early years at secondary school.
OK, let me solve it as follows:
Firstly, to be clear about notation, when I write ABC, it means a 3-digit integer, such as 235, 536, and so on.
It is remarked that:
ABC=100*A+10*B+C=100*(A+B/10+C/100)

Statement 1 is equevalent to ABC=BCA, therefore A=B=C, or ABC=111, 222, ..., 999
The same for statement 2.

On the other hand, 111, 222, ..., 999 are multiples of 37.
Therefore correct answer is D.
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Re: The Quest for 700: Weekly GMAT Challenge   [#permalink] 27 Jul 2010, 03:16
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