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 Post subject: GMAT Number Theory (Data Sufficiency)
PostPosted: Thu Aug 05, 2010 6:46 am 
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The sum of the digits of a three-digit number is 11. What is the product of the three digits?
(1) The number is divisible by 5.
(2) The hundreds digit is twice the tens digit.

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) alone is not sufficient because it tells us only that the number has a units digit of 0 or 5. If the number has a units digit of 0, the product of the digits will be 0 (since 0 times any number is 0), and if the number has a units digit of 5, the product of the digits might not be zero (depending on whether one of the other digits is 0). For instance, the number 515 has digits that add to 11, yet the product of the digits is 25. Therefore, Statement (1) is insufficient.

Statement (2) is not sufficient by itself either. The number could be 218 or 425 or 632. The product of the digits is different in each case.

Combined, the two statements are sufficient. The only possibility is that the number is 425, and the product of the digits is 40.

Since the statements are both insufficient individually but sufficient when combined, the correct answer is choice (C).
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Question never says that the number has distict digits. But, the solution assumes this.


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 Post subject: Re: math (test 4, question 3): number theory, data sufficien
PostPosted: Thu Aug 05, 2010 7:05 am 
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Joined: Sun May 30, 2010 2:23 am
Posts: 498
The solution doesn't assume that digits must be distinct. However when we use both statements and find the only possible number - it turns out to have all distinct digits.

Let us go over the solution once again:

When we use statement (1) we pick at least two numbers, showing that both numbers satisfy criteria but answer question differently:
515: 5 × 1 × 5 = 25
560: 5 × 6 × 0 = 0

When we use statement (2) we also pick at least two numbers, showing that both numbers satisfy criteria but answer question differently:
218: 2 × 1 × 8 = 16
425: 4 × 2 × 5 = 40

When we use both statements we start with the second one "(2) The hundreds digit is twice the tens digit." So the tens digit can not exceed 4 because otherwise the hundreds digit would not be a digit (it would be greater than 9). So we have only 4 possible options for first two digits:
21_
42_
63_
84_

If use original statement, "The sum of the digits of a three-digit number is 11.", we can find the units digit in each case:
218: 2 + 1 + 8 = 11
425: 4 + 2 + 5 = 11
632: 6 + 3 + 2 = 11
8 + 4 = 12 > 11 so this choice is not possible.

When we also use statement (1),"The number is divisible by 5." , we see that only 1 number out of three fits all the criteria, 425, so in this case we can give a definite answer. Both statements combined are sufficient.

Note, that the key for solving this question is proper use of statement (2) and we never assume that digits must be distinct.


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