GMAT Math : Data-Sufficiency Questions
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Example Questions
Example Question #1001 : Data Sufficiency Questions
Let A-B = C. If B is an integer not equal to 0, is C an integer?
1. A/B is an integer
2. A*B is an integer
Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.
Each statement alone is sufficient.
Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.
Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.
Statements 1 and 2 together are not sufficient.
Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.
Let's first look at what the question is asking for. They want us to determine if C is an integer. Since we know that B is an integer, C will be an integer only if A is an integer. If A not an integer, C will not be an integer. So from statements 1 and 2, we want to see if we can prove if A is definitely an integer.
First, let's try statement 2, which says that A*B is an integer. Let's see what happens if B is 2. If B is 2, then 2.5*2 = 5 is an integer, but 2*2 = 4 is also an integer. So A in this case could be an integer or a not. So statement 2 alone is not sufficient to get our answer.
Now, let's go back to statement 1, which says A/B is an integer. Let's name another variable x. Let x = A/B. If x=A/B we can rewrite this as A = x*B. But since we know that x is an integer (given in the statement) and B is a non-zero integer (given in the question), A is therefore an integer! (The product of two integers is ALWAYS an integer.) This statement alone is enough to prove that A is an integer.
Since, as discussed before, A is definitely an integer, and B is an integer. An integer minus an integer will always be an integer. Therefore C is an integer, and statement 1 is sufficient to answer the question.
Example Question #1002 : Data Sufficiency Questions
Statement 1: 
Statement 2: 
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 alone provides insufficient information to answer the question; for example, 
Statement 2, however, provides proof that 
Example Question #3111 : Gmat Quantitative Reasoning
Let 
Statement 1:
Statement 2: 
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
If we know Statement 1, we can rewrite 
If we know Statement 2, we have a system of linear equations that we can solve to get 
By susbstituting: 
Either statement is sufficient to prove 
Example Question #1003 : Data Sufficiency Questions
Does the integer 
(1) 
(2) 
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
BOTH statements TOGETHER are not sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.
EACH statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(1) This statement tells us that 2 and 7 are prime factors of 
(2) This statement tells us that 3 and 5 are prime factors of 
Considering both (1) and (2), we have four positive prime factors of 
Example Question #1004 : Data Sufficiency Questions
What is the remainder when the two digit, positive integer 
(1) The sum of the digits of 
(2) The difference of the digits is 3.
D. EACH statement ALONE is sufficient.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
For statement (1), there are 3 possible two digit positive integers: 30, 12, 21. The remainder when these numbers are divided by 3 is 0. Therefore, statement (1) is sufficient. For statement (2), there are lots of different combinations of integers that have different remainders when divided by 3. For example, the remainder of 30 is 0, but the remainder of 14 is 2. Therefore, statement (2) is insufficient.
Example Question #1005 : Data Sufficiency Questions
(1) 
(2) 
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1) tells us that 
Statement (2) tells us that 
Example Question #3111 : Gmat Quantitative Reasoning
Is a whole number a perfect square integer?
Statement 1: It falls between 110 and 120 inclusive.
Statement 2: Its last digit is a 7.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
There are no perfect square integers between 110 and 120 inclusive:
Also, all perfect squares end in 0, 1, 4, 5, 6, or 9, depending on the last digit of the number being squared.
From ether statement alone, it therefore follows that the number is not a perfect square integer.
Example Question #3121 : Gmat Quantitative Reasoning
Is a whole number a perfect square?
Statement 1: Its last digit is 9.
Statement 2: It falls between 160 and 170 inclusive.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 alone does not answer the question, since some integers ending in 9 are perfect squares (9, 49) and some are not (29, 39).
Statement 2 alone does not answer the question, since of the numbers in that range, only 169 is a perfect square (of 13).
From both statements together, however, it follows that the number in question is 169, which, as just stated, is the square of 13.
Example Question #3122 : Gmat Quantitative Reasoning
I am thinking of two distinct prime numbers. What are they?
Statement 1: Both are between 50 and 60.
Statement 2: Their sum is 112.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 1 is a giveaway; the only prime numbers between 50 and 60 are 53 and 59.
Statement 2 is not a giveaway, though, since there are at least two sums of primes equal to 112:
Example Question #1010 : Data Sufficiency Questions
Is a positive integer 
Statement 1:
Statement 2:
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
From Statement 1 it follows that 
If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that 
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