Test: Number Properties - GMAT MCQ

Let's analyze each statement separately:

Statement (1): x - p = q - x = k, where p, q, and k are prime numbers.

This statement provides an equation involving x and prime numbers p, q, and k. However, it doesn't give us any specific information about the value of x or whether x itself is a prime number. Therefore, statement (1) alone is not sufficient to determine whether x is a prime number.

Statement (2): The total odd factor of 15k³ is 4, where k is a prime number.

This statement provides information about the total odd factors of a specific expression involving the prime number k. However, it doesn't provide any direct information about the value of x or whether x is a prime number. Therefore, statement (2) alone is not sufficient to determine whether x is a prime number.

Now let's consider both statements together. Even when considering both statements together, we still don't have any direct information about the value of x or whether x is a prime number. The statements provide information about prime numbers in different contexts, but they don't provide any direct information about x itself.

Therefore, when we consider both statements together, we still cannot determine whether x is a prime number. 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): The sum of all the numbers in Q is zero.

This statement tells us that the sum of all the numbers in Q is zero. However, it doesn't provide any information about the individual numbers in Q or whether they are equal. For example, Q could contain both positive and negative numbers that cancel each other out, resulting in a sum of zero. Therefore, statement (1) alone is not sufficient to determine whether all the numbers in Q are equal.

Statement (2): The product of any two numbers in Q is zero.

This statement tells us that the product of any two numbers in Q is zero. This means that at least one of the numbers in each pair is zero. However, it doesn't provide any information about the other numbers in Q or whether they are equal. Therefore, statement (2) alone is not sufficient to determine whether all the numbers in Q are equal.

Now let's consider both statements together. From statement (1), we know that the sum of all the numbers in Q is zero. From statement (2), we know that the product of any two numbers in Q is zero, which means at least one number in each pair is zero.

When we consider both statements together, we can conclude that all the numbers in Q must be equal to zero. If there were any non-zero numbers in Q, their sum would not be zero, and their product would not be zero either. Therefore, the only possibility for all the numbers in Q to satisfy both conditions is if they are all zero.

Therefore, both statements together are sufficient to determine that all the numbers in Q are equal. Hence, the answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): x is an integer.
If x is an integer, and we know that x + y is an integer, we can conclude that y must also be an integer. The sum of two integers is always an integer. Therefore, statement (1) alone is sufficient to answer the question.

Statement (2): x + 2y is an integer.
Similar to the reasoning above, if x + 2y is an integer, and x is an integer, we can deduce that 2y must also be an integer. Since 2y is an integer, y itself doesn't have to be an integer. For example, if x = 1 and y = 0.5, then x + 2y = 2, which is an integer. Therefore, statement (2) alone is not sufficient to answer the question.

However, when we consider both statements together, we can conclude that y is an integer. Statement (1) tells us that x is an integer, and statement (2) tells us that x + 2y is an integer. From these two pieces of information, we can determine that y must be an integer in order for the sum x + y to be an integer.

Therefore, each statement alone is sufficient to answer the question, and the answer is D.

To find the value of k, we need to determine its prime factorization based on the given information. We know that k has exactly two prime factors: 3 and 7. Therefore, we can express k as:

k = 3^a * 7^b

where a and b are positive integers representing the exponents.

Now, let's consider the number of positive factors of k. The total number of factors of a number can be found by multiplying the exponents of each prime factor by one more than each exponent, and then multiplying those results together.

In this case, the number of factors of k is given as 6. So we have:

(a + 1) * (b + 1) = 6

To determine the value of k, we need to find the values of a and b.

Now let's analyze the given statements:

Statement (1): 32 is a factor of k.
This implies that k must be divisible by 32, which is equal to 2⁵. Since k only has prime factors of 3 and 7, it must also have 2 as a prime factor. Therefore, a must be at least 5 in order for k to be divisible by 32. With a = 5, we have:

(5 + 1) * (b + 1) = 6
6 * (b + 1) = 6
b + 1 = 1
b = 0

So, k = 3⁵ * 7⁰ = 3⁵ = 243.

Statement (1) alone is sufficient to determine the value of k.

Statement (2): 72 is NOT a factor of k.
This statement doesn't provide direct information about the value of k. It only tells us that k is not divisible by 72. However, we already have sufficient information from statement (1) to determine the value of k.

Therefore, statement (2) alone is not sufficient to determine the value of k.

Since statement (1) alone is sufficient to determine k, the correct answer is:

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

To solve this problem, let's consider the possible remainders when dividing consecutive positive integers greater than 9 by 5.

When dividing a positive integer by 5, the possible remainders are 0, 1, 2, 3, or 4.

Now let's analyze the given statements:

Statement (1): The sum of the remainders is even.
This means that the sum of the remainders when dividing the two consecutive integers by 5 is an even number. Let's consider the possible cases:

Case 1: Remainders are 0 and 0
In this case, the sum of the remainders is 0 + 0 = 0, which is even.

Case 2: Remainders are 1 and 2
In this case, the sum of the remainders is 1 + 2 = 3, which is odd.

Case 3: Remainders are 2 and 3
In this case, the sum of the remainders is 2 + 3 = 5, which is odd.

Case 4: Remainders are 3 and 4
In this case, the sum of the remainders is 3 + 4 = 7, which is odd.

Therefore, the only case that satisfies the statement is Case 1, where the remainders are both 0. This means that the two consecutive integers are multiples of 5.

Statement (1) alone is sufficient to determine that the sum of the remainders is 0, an even number.

Statement (2): The sum of the units digits of the two original integers is 9.
This statement provides information about the units digits of the two integers, but it does not directly give any information about the remainders when divided by 5. It is possible to have multiple sets of consecutive integers with a sum of units digits equal to 9, and the remainder when dividing them by 5 can vary.

Statement (2) alone is not sufficient to determine the sum of the remainders.

Since statement (1) alone is sufficient to determine that the sum of the remainders is even, and statement (2) does not provide enough information, the correct answer is:

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

Statement (1): x is not prime.
This statement tells us that x is a composite number, meaning it has factors other than 1 and itself. If x is composite, we can find the factors of x and calculate their sum to obtain y. However, this statement alone doesn't provide information about the specific values of x and y or whether |x - y| > 1.

For example, if x = 6, which is not prime, the factors of 6 are 1, 2, 3, and 6. The sum of these factors is y = 1 + 2 + 3 + 6 = 12. In this case, |x - y| = |6 - 12| = 6 > 1.

On the other hand, if x = 4, which is also not prime, the factors of 4 are 1, 2, and 4. The sum of these factors is y = 1 + 2 + 4 = 7. In this case, |x - y| = |4 - 7| = 3 > 1.

Statement (1) alone is not sufficient to determine whether |x - y| > 1.

Statement (2): x does not equal 1.
This statement simply tells us that x is not equal to 1. While this implies that x is a positive integer greater than 1, it doesn't provide any direct information about the factors of x, the sum of the factors, or the difference |x - y|.

For example, if x = 6, the factors of 6 are 1, 2, 3, and 6. The sum of these factors is y = 1 + 2 + 3 + 6 = 12. In this case, |x - y| = |6 - 12| = 6 > 1.

On the other hand, if x = 2, the factors of 2 are 1 and 2. The sum of these factors is y = 1 + 2 = 3. In this case, |x - y| = |2 - 3| = 1 ≤ 1.

Statement (2) alone is not sufficient to determine whether |x - y| > 1.

When considering both statements together, we have the following information:

From statement (1): x is not prime.
From statement (2): x does not equal 1.

These statements together imply that x is a composite number greater than 1. In other words, x has factors other than 1 and itself. Since y is the sum of all the unique factors of x, y will always be greater than 1.

Therefore, |x - y| will always be greater than 1.

When considering both statements together, we can determine that |x - y| > 1.

Hence, both statements together are sufficient to answer the question asked.

The correct answer is:

C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): The remainder is 3 when n is divided by 8.
This statement provides information about the remainder when n is divided by 8, but it does not directly give any information about the remainder when n is divided by 6.

For example, if n is 11, the remainder when dividing by 8 is 3, but the remainder when dividing by 6 is 5.

Statement (1) alone is not sufficient to determine the remainder when n is divided by 6.

Statement (2): The remainder is 8 when n-1 is divided by 9.
This statement provides information about the remainder when n-1 is divided by 9, but it does not directly give any information about the remainder when n is divided by 6.

For example, if n is 18, the remainder when dividing n-1 (17) by 9 is 8, but the remainder when dividing n by 6 is 0.

Statement (2) alone is not sufficient to determine the remainder when n is divided by 6.

When considering both statements together, we have the following information:

From statement (1): The remainder is 3 when n is divided by 8.
From statement (2): The remainder is 8 when n-1 is divided by 9.

Let's analyze this information further:

If the remainder when n is divided by 8 is 3, it means that n can be expressed as 8k + 3, where k is an integer.

If the remainder when n-1 is divided by 9 is 8, it means that n-1 can be expressed as 9m + 8, where m is an integer.

From the expressions above, we can deduce that n-1 is divisible by both 8 and 9.

To find the possible values of n that satisfy both statements, we can consider the multiples of the least common multiple (LCM) of 8 and 9, which is 72.

The multiples of 72 can be expressed as 72t, where t is an integer.

To satisfy the condition n-1 is divisible by 72, we have:

9m + 8 = 72t

Rearranging the equation, we get:

9m = 72t - 8
9m = 8(9t - 1)

The right side of the equation must be divisible by 9, so 9t - 1 must be divisible by 9. This means t must be 1, 10, 19, 28, and so on.

For t = 1, we have:

9m = 8(9(1) - 1)
9m = 8(8)
9m = 64
m = 7

Therefore, one possible value of n is 72t + 1 = 72(1) + 1 = 73.

However, we have multiple possible values of t that satisfy the condition, such as t = 10, 19, 28, and so on. These values would give us different values of n.

Since there are multiple possible values of n that satisfy both statements and lead to different remainders when divided by 6, we cannot definitively determine the remainder when n is divided by 6 based on the given information.

Therefore, the correct answer is:

C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): x has no prime multiples.
This statement implies that x is not divisible by any prime number other than 1 and itself. If x has no prime multiples, it means that x is not divisible by 2 or any other prime number. However, this information does not directly tell us whether x is equal to 3 or not.

For example, x could be any other integer that is not divisible by any prime number. It could be 5, 7, 11, or any other non-prime number.

Statement (1) alone is not sufficient to determine whether x is equal to 3 or not.

Statement (2): x has no odd multiples.
This statement implies that x is not divisible by any odd number other than 1 and itself. If x has no odd multiples, it means that x is not divisible by 3 or any other odd number. However, this information does not directly tell us whether x is equal to 3 or not.

For example, x could be any other even integer or any integer that is not divisible by any odd number. It could be 2, 4, 6, or any other non-odd number.

Statement (2) alone is not sufficient to determine whether x is equal to 3 or not.

When considering both statements together, we can deduce the following:

If x has no prime multiples (statement 1) and no odd multiples (statement 2), it means that x is not divisible by any prime number or any odd number other than 1 and itself. In other words, x is not divisible by any number other than 1 and itself. This is the definition of a prime number.

Since 3 is a prime number, if x satisfies both statements 1 and 2, it means that x must be equal to 3.

Therefore, when considering both statements together, we can determine that x is equal to 3.

Hence, each statement alone is sufficient to answer the question asked.

The correct answer is:

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

Statement (1): The sum of a and b is prime.
If a and b are prime numbers, their sum will only be prime if one of the numbers is 2 (the only even prime number) and the other number is odd. In this case, ab will be even since one of the factors (2) is even.

However, if both a and b are odd prime numbers, their sum will be even, but the product ab will also be odd.

Therefore, statement (1) alone is sufficient to determine that ab is even if the sum of a and b is prime, but it does not definitively determine that ab is even if the sum of a and b is not prime.

Statement (2): The difference of a and b is prime.
The difference between two prime numbers can be either even or odd. For example, if a = 5 and b = 2, their difference is 5 - 2 = 3, which is prime. In this case, ab = 5 * 2 = 10, which is even.

However, if a = 5 and b = 3, their difference is 5 - 3 = 2, which is prime. In this case, ab = 5 * 3 = 15, which is odd.

Therefore, statement (2) alone is not sufficient to determine whether ab is even or not.

Since statement (1) alone is sufficient to determine that ab is even if the sum of a and b is prime, but statement (2) does not provide enough information, the correct answer is:

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

Statement (1): 3n is even.
If 3n is even, it means that 3n is divisible by 2. Dividing both sides of the equation by 3, we have n = (2k)/3, where k is an integer.

From this equation, we can see that n can be expressed as a fraction. Therefore, statement (1) alone is not sufficient to determine whether n is an even integer.

Statement (2): 5n is even.
Similar to statement (1), if 5n is even, it means that 5n is divisible by 2. Dividing both sides of the equation by 5, we have n = (2k)/5, where k is an integer.

Again, n can be expressed as a fraction, so statement (2) alone is not sufficient to determine whether n is an even integer.

When considering both statements together, we have the following information:

n = (2k)/3 (from statement 1)
n = (2k)/5 (from statement 2)

For n to be an integer, both k/3 and k/5 must be integers. This implies that k must be divisible by both 3 and 5, which means k must be a multiple of 15.

Considering k = 15, we have:

n = (2 * 15)/3 = 10

n is an even integer in this case.

However, considering k = 30, we have:

n = (2 * 30)/3 = 20

n is an even integer in this case as well.

Since there are multiple values of k that satisfy both statements and lead to n being an even integer, we cannot definitively determine whether n is an even integer or not based on the given information.

Therefore, the correct answer is:

C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.