Test: Number Properties - GMAT MCQ

Statement (1): All students come from three colleges, X, Y, and Z, that sent 12, 10, and 9 students, respectively.
From this statement alone, we know the number of students from each college, which are 12, 10, and 9. Since all students from the same college need to sit together, we can treat each group of students from the same college as a single entity. Therefore, we have three groups: X, Y, and Z. The number of arrangements can be calculated by treating these three groups as distinct entities and permuting them. The maximum number of arrangements would be (12!)(10!)(9!).

Statement (2): N is a prime number between 30 and 40.
From this statement alone, we know that the value of N is a prime number between 30 and 40. However, we don't have any information about the distribution of students among the colleges or the number of students from each college. Without knowing these details, we cannot determine the maximum number of arrangements. Statement (2) alone is not sufficient.

Combining both statements, we have the following information:

The number of students from each college: 12, 10, and 9 (Statement 1).
The value of N is a prime number between 30 and 40 (Statement 2).
Using Statement 1, we can determine the maximum number of arrangements by considering the three groups as distinct entities and permuting them. However, Statement 2 does not provide any additional information about the distribution or arrangement of the students. Therefore, Statement 1 alone is sufficient to answer the question, but Statement 2 alone is not sufficient.

Hence, the correct answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): z³ - 3 is an odd integer.
If z³ - 3 is an odd integer, it means that z³ is an even integer because subtracting an odd integer from an even integer always results in an odd integer. For z³ to be an even integer, z must also be an even integer. This implies that z itself is an even integer. Therefore, statement (1) alone is sufficient to conclude that z is not an odd integer.

Statement (2): z - 3 is an odd integer.
If z - 3 is an odd integer, it means that z is an even integer because subtracting an odd integer from an even integer always results in an odd integer. Therefore, statement (2) alone is sufficient to conclude that z is not an odd integer.

Both statements (1) and (2) individually provide sufficient information to determine that z is not an odd integer. Therefore, each statement alone is sufficient to answer the question asked.

Hence, the answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1): n = 2x + 1, where x is a positive integer.
This statement provides information about the form of n, indicating that n is an odd integer. However, knowing that n is odd doesn't provide enough specific information to determine the units digit of 3n. For example, if n = 3, then 3n = 3 * 3 = 9, and the units digit is 9. But if n = 5, then 3n = 3 * 5 = 15, and the units digit is 5. The units digit of 3n can vary depending on the specific value of n. Statement (1) alone is not sufficient.

Statement (2): n = 2k - 1, where k is a positive integer.
This statement also provides information about the form of n, indicating that n is an odd integer. Similar to statement (1), knowing that n is odd doesn't provide enough specific information to determine the units digit of 3n. For example, if n = 1, then 3n = 3 * 1 = 3, and the units digit is 3. But if n = 3, then 3n = 3 * 3 = 9, and the units digit is 9. Again, the units digit of 3n can vary depending on the specific value of n. Statement (2) alone is not sufficient.

When we consider both statements together, we still don't have enough information to determine the units digit of 3n. Both statements provide similar information about the form of n, but they don't give any specific values for n that would allow us to determine the units digit of 3n uniquely. Therefore, even when considering both statements together, we still need additional data to answer the question.

Hence, the answer is E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): A and B both have the same Even/Odd nature.
This statement tells us that A and B have the same parity, meaning they are both either even or odd. If A and B have the same parity, subtracting 7B from A will not change their parity. For example, if A and B are both even, then 7B will also be even, and subtracting an even number from an even number results in an even number. The same holds true if A and B are both odd. Therefore, statement (1) alone is sufficient to determine if A - 7B is even.

Statement (2): A and B are both divisible by 100.
This statement tells us that both A and B are multiples of 100. However, being divisible by 100 does not provide any specific information about the parity of A or B. For example, A could be an even multiple of 100 and B could be an odd multiple of 100, or vice versa. Without knowing the parity of A and B, we cannot determine the parity of A - 7B. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know that A and B have the same parity (statement 1) and both are divisible by 100 (statement 2). This means that A and B are both even numbers. Subtracting an even number (7B) from another even number (A) will result in an even number. Therefore, when considering both statements together, we can conclude that A - 7B is even.

Hence, the answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1): x + y = 98.
This statement tells us that the sum of x and y is 98. Since x and y are consecutive even integers, they can be represented as x and x+2 (where x is the smaller even integer). Therefore, we have the equation x + (x+2) = 98. By solving this equation, we can determine the value of x and consequently y, which allows us to calculate xy. Statement (1) alone is sufficient.

Statement (2): y - x = 2.
This statement tells us that the difference between y and x is 2. However, knowing this difference alone doesn't provide enough information to determine the values of x and y uniquely. There could be multiple pairs of consecutive even integers with a difference of 2 between them. For example, x = 4 and y = 6 satisfy the equation, as well as x = 8 and y = 10. Without additional information, we cannot determine the specific values of x and y, and thus, we cannot determine the value of xy. Statement (2) alone is not sufficient.

When we consider both statements together, statement (1) tells us that x + y = 98, and statement (2) tells us that y - x = 2. By solving these two equations simultaneously, we can determine the unique values of x and y. For example, by adding equation (1) and equation (2), we get 2y = 100, which implies y = 50. Substituting this value into equation (2), we find x = 48. Thus, we can determine the specific values of x and y and calculate xy. When considering both statements together, we can answer the question.

Therefore, the answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): The tens digit of y is 3.
This statement provides some information about y, but it doesn't give any specific information about the hundreds or units digit of y, which are important for comparing x and y. Alone, this statement is not sufficient.

Statement (2): The rightmost digit of x is 1, and the rightmost digit of y is 9.
This statement gives specific information about the rightmost digits of both x and y. Since x is obtained by reversing the digits of y, the rightmost digits of x and y will be the same. Therefore, x must also end with 9. As a result, we can conclude that x is greater than y. Alone, this statement is sufficient to answer the question.

Since statement (2) alone is sufficient to answer the question, but statement (1) alone is not, the answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Given info →
p and m are integers.
We are asked if m is multiple of 6 or not.

Statement 1 →
if p = 1 ⇒ m is divisible by 6
if p = 2 ⇒ m is not divisible by 6

hence not sufficient

Statement 2 →
m = 7! + 3
m = 6k+3 for some integer k
Hence it will leave a remainder 3 when divide by 6.
Hence m is not divisible by 6.

Hence Sufficient

Hence B

Statement 1: x = 5
We don't know where the sequence starts. For example:
Case a: the 5 numbers are {2, 4, 6, 8, 10}, in which case the sum = 30
Case b: the 5 numbers are {4, 6, 8, 10, 12}, in which case the sum = 40
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The least integer in the sequence is 6
We don't know how many integers are in the sequence. For example:
Case a: the numbers are {6, 8}, in which case the sum = 14
Case b: the numbers are {6, 8, 10}, in which case the sum = 24
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
We now know that the numbers are {6, 8, 10, 12, 14}, in which case the sum = 50
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Statement (1): x is the square root of an integer.
From this statement alone, we can deduce that x is a positive or negative integer, or zero. However, we don't have any information about whether x is odd or even. For example, x could be the square root of 4, which is an even integer, or it could be the square root of 9, which is an odd integer. Therefore, statement (1) alone is not sufficient to determine if x is an odd integer.

Statement (2): x is the square of an integer.
From this statement alone, we can determine that x must be a non-negative integer (including zero). However, we still don't have any information about whether x is odd or even. For example, x could be the square of 2, which is an even integer, or it could be the square of 3, which is an odd integer. Therefore, statement (2) alone is not sufficient to determine if x is an odd integer.

Since neither statement alone provides enough information to determine if x is an odd integer, and the two statements together don't provide any additional information, we need additional data to answer the question. Hence, the correct answer is E: Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

Statement (1): K = 3M where M is an integer.
Statement (2): K = 6J where J is an integer.

From Statement (1), we know that K is a multiple of 3. However, this does not provide any information about divisibility by 2, which is necessary to determine if K is odd.

From Statement (2), we know that K is a multiple of 6. Since 6 is divisible by 2, any multiple of 6 will also be divisible by 2. Therefore, we can conclude that K is even based on Statement (2) alone.

Since Statement (2) alone is sufficient to determine that K is even, but Statement (1) does not provide enough information, the answer is option B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient to answer the question asked.