Test: Divisibility/Multiples/Factors - GMAT MCQ

Statement (1): a is a factor of b.

If a is a factor of b, then ak will be a factor of bk for any positive integer k. Therefore, statement (1) alone is sufficient to answer the question.

Statement (2): k = m.

If k = m, then ak will be a factor of bm for any positive integer a and b. Therefore, statement (2) alone is sufficient to answer the question.

When we consider both statements together, we have the information that a is a factor of b and k = m. Combining these conditions, we can conclude that ak is a factor of bm. Therefore, both statements together are also sufficient to answer the question.

Since both statements alone are sufficient, and together they are also sufficient, 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): 24k is divisible by 20.

If 24k is divisible by 20, it means that k is divisible by 20/24, which simplifies to 5/6. However, this does not necessarily mean that k is divisible by 5. For example, if k = 10, then 24k = 240, which is divisible by 20 but not by 5. Therefore, statement (1) alone is not sufficient to determine if k is divisible by 5.

Statement (2): k + 1 is divisible by 6.

If k + 1 is divisible by 6, it means that k is one less than a multiple of 6. In other words, k = 6n - 1 for some integer n. This does not provide any information about whether k is divisible by 5. For example, if k = 5, then k + 1 = 6, which is divisible by 6 but not by 5. Therefore, statement (2) alone is not sufficient to determine if k is divisible by 5.

Combining both statements, we have some information about the divisibility of k, but it is still not enough to determine if k is divisible by 5. Therefore, both statements together are not sufficient.

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

Statement (1): x is divisible by 6.

If x is divisible by 6, it means that x can be written as x = 6k for some integer k. Adding y to x, we have x + y = 6k + y. From this equation, we can see that whether x + y is divisible by 6 depends on the value of y. Statement (1) alone does not provide any information about the divisibility of y, so it is not sufficient to determine if x + y is divisible by 6.

Statement (2): y is divisible by 6.

If y is divisible by 6, it means that y can be written as y = 6m for some integer m. Adding x to y, we have x + y = x + 6m. Similarly to statement (1), statement (2) alone does not provide any information about the divisibility of x, so it is not sufficient to determine if x + y is divisible by 6.

When we consider both statements together, we know that x is divisible by 6 (statement 1) and y is divisible by 6 (statement 2). In this case, x can be written as x = 6k and y can be written as y = 6m, where k and m are integers. Adding x and y, we have x + y = 6k + 6m = 6(k + m). Since k + m is an integer, we can conclude that x + y is divisible by 6.

Therefore, together, statements (1) and (2) are sufficient to answer the question. 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): p is a multiple of 2.

If p is a multiple of 2, it means that p is an even number. However, being an even number does not necessarily imply that p is a multiple of 24. For example, p could be 2 or 6, which are multiples of 2 but not multiples of 24. Statement (1) alone is not sufficient to determine if p is a multiple of 24.

Statement (2): p is a multiple of 12.

If p is a multiple of 12, it means that p can be written as p = 12k for some integer k. Since 12 is a multiple of 24, any multiple of 12 is also a multiple of 24. Therefore, statement (2) alone is sufficient to determine that p is a multiple of 24.

When we consider both statements together, we know that p is a multiple of 2 (statement 1) and p is a multiple of 12 (statement 2). However, being a multiple of 2 and a multiple of 12 does not guarantee that p is specifically a multiple of 24. For example, p could be 6, which satisfies both statements but is not a multiple of 24. Therefore, even when considering both statements together, we cannot definitively conclude that p is a multiple of 24.

Since we cannot answer the question with certainty using the given information, the answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): n - 3 = 2k

From this equation, we can rewrite it as n = 2k + 3. We can see that n is expressed in terms of k, but we don't have any information about whether n is divisible by 7. Statement (1) alone is not sufficient to determine if n is divisible by 7.

Statement (2): 2k - 4 is divisible by 7

If 2k - 4 is divisible by 7, it means that 2k - 4 = 7m for some integer m. Rearranging the equation, we have 2k = 7m + 4, which implies that 2k is of the form 7m + 4. This tells us that 2k has a remainder of 4 when divided by 7, but it doesn't provide any information about whether n is divisible by 7. Statement (2) alone is not sufficient to determine if n is divisible by 7.

When we consider both statements together, we have the equations n = 2k + 3 (from statement 1) and 2k = 7m + 4 (from statement 2). However, even with both equations, we cannot definitively determine if n is divisible by 7. The two equations provide independent information about k and n, but they don't allow us to determine the divisibility of n by 7.

Therefore, 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): n is the product of two different prime numbers.

If n is the product of two different prime numbers, then n can be expressed as the product of p and q, where p and q are distinct prime numbers. In this case, n has two distinct prime factors. Since n is an integer, it must also be divisible by 1 and itself. Therefore, n is divisible by at least four positive integers.

Statement (2): n and 2³ are each divisible by the same number of positive integers.

This statement tells us that the number of positive divisors of n is the same as the number of positive divisors of 2³. Since 2³ is a prime number, it has exactly two positive divisors: 1 and 2³. If n has the same number of positive divisors as 2³, it means that n is also a prime number and has exactly two positive divisors: 1 and itself. Therefore, n is divisible by exactly two positive integers.

From the evaluation of both statements, we can see that each statement alone is sufficient to answer the question. Statement (1) tells us that n is divisible by at least four positive integers, and Statement (2) tells us that n is divisible by exactly two positive integers. Therefore, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

Let's consider the factors of B. Since B has only two distinct factors, they must be 1 and B itself. The difference between these factors is B - 1.

Given that the difference between the two distinct factors is odd, we can conclude that B - 1 is an odd number. Therefore, B must be an even number.

To find the value of B+1, we add 1 to the even number B. Adding 1 to an even number always results in an odd number. Therefore, B+1 will be an odd number.

Among the answer choices, the only odd number is 3. Hence, the correct answer is (B) 3

Statement (1): m is divisible by 6.

Since x/n = m, we can rewrite this equation as x = mn. From this equation, we can see that x will be divisible by 6 if both m and n are divisible by 6. However, statement (1) only tells us that m is divisible by 6 and provides no information about the divisibility of n. Therefore, statement (1) alone is not sufficient to determine if x is divisible by 3.

Statement (2): n is divisible by 15.

Similar to the previous analysis, we know that x will be divisible by 3 if both m and n are divisible by 3. However, statement (2) only tells us that n is divisible by 15 and provides no information about the divisibility of m. Therefore, statement (2) alone is not sufficient to determine if x is divisible by 3.

Since statement (1) tells us that m is divisible by 6 and statement (2) tells us that n is divisible by 15, we can conclude that both m and n are divisible by their least common multiple (LCM), which is 30. Therefore, we can express x as x = 30k, where k is an integer.

Since 30 is divisible by 3, we can conclude that x is also divisible by 3. Therefore, each statement alone (statement (1) and statement (2)) is individually sufficient to determine that x is divisible by 3.

Based on the analysis above, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

Statement (1): n is a multiple of 20.

If n is a multiple of 20, it means that n can be written as 20k, where k is an integer. However, this information does not provide any direct information about whether n is a multiple of 15 or not. For example, if k = 1, then n = 20, which is not a multiple of 15. If k = 3, then n = 60, which is a multiple of 15. Therefore, statement (1) alone is not sufficient to determine if n is a multiple of 15.

Statement (2): n + 6 is a multiple of 3.

If n + 6 is a multiple of 3, it means that n + 6 can be written as 3m, where m is an integer. Rewriting this equation, we have n = 3m - 6. From this equation, we can see that n will be divisible by 3 if it is a multiple of 3 minus 6, which is equivalent to a multiple of -3. Therefore, statement (2) alone is not sufficient to determine if n is a multiple of 15.

Combining both statements, we know that n is a multiple of 20 (statement 1) and n is a multiple of -3 (statement 2). Since 20 is a multiple of 4 and -3 is not, the common multiple of 20 and -3 is their least common multiple (LCM), which is 60. Therefore, n must be a multiple of 60. In particular, n can be expressed as n = 60k, where k is an integer.

Since 60 is a multiple of 15, we can conclude that n is also a multiple of 15. Therefore, both statements together are sufficient to determine that n is a multiple of 15.

Based on the analysis above, 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): 7 is a factor of n.

If 7 is a factor of n, it means that n is divisible by 7. However, this does not provide any information about the specific value of n. For example, n could be 7, 14, 21, and so on. Therefore, statement (1) alone is not sufficient to determine the value of n.

Statement (2): n is a prime number.

If n is a prime number, it means that n is only divisible by 1 and itself. This tells us that n cannot have any factors other than 1 and n. However, it does not provide any information about whether 7 is a factor of n or not. For example, if n = 5, it is a prime number but not divisible by 7. Therefore, statement (2) alone is not sufficient to determine the value of n.

Combining both statements, we know that n has 7 as a factor and that n is a prime number. This means that n must be a multiple of 7 and cannot have any other factors besides 1 and itself. However, this still does not give us enough information to determine the specific value of n. For example, n could be 7, 14, 21, and so on. Therefore, both statements together are not sufficient.

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