Test: Divisibility/Multiples/Factors - GMAT MCQ

Factorize: 24x = 2³ ∗ 3 ∗ x

(1) x is a two-digit number
Clearly insufficient.
(2) x² has 3 positive factors

The above means that x is a prime number. Only the squares of primes (p²) have three factors: 1, p, and p²
If x = 11 (or any other prime except 2 and 3), then 24x = 2³ ∗ 3 ∗ 11 will have (3+1)(1+1)(1+1) = 16
factors;
If x = 2, then 24x = 2⁴ ∗ 3 will have (4 + 1)(1 + 1)=10 factors.
If x = 2, then 24x = 2³ ∗ 3² will have (3 + 1)(2 + 1) = 12 factors.
Not sufficient.

(1)+(2) Since (1) says that x is a two-digit number, then x cannot be 2 or 3, so it must be a two-digit prime. For any two digit prime, 24x = 2³ ∗ 3 ∗ x will have (3 + 1) (1 + 1) (1 + 1) =16 factors. Sufficient.

Statement (1): 'a' is not divisible by 7.
This statement alone does not provide any information about 'b.' It only tells us about 'a.' Therefore, we cannot determine if the sum of 'a' and 'b' is divisible by 7 based on this statement alone.

Statement (2): 'a - b' is divisible by 7.
This statement provides a relationship between 'a' and 'b.' If 'a - b' is divisible by 7, it implies that the difference between 'a' and 'b' is a multiple of 7. However, this does not necessarily mean that the sum of 'a' and 'b' is divisible by 7.

When we consider both statements together, we can gain some insights. Since 'a - b' is divisible by 7, we know that the difference between 'a' and 'b' is a multiple of 7. This implies that 'a' and 'b' have the same remainder when divided by 7. If 'a' is not divisible by 7, it means it has a remainder. Therefore, 'b' must also have the same remainder when divided by 7.

Now, let's consider a scenario where 'a' has a remainder of 1 when divided by 7. In this case, 'b' would also have to have a remainder of 1 when divided by 7 to satisfy statement (2). The sum of two numbers with a remainder of 1 when divided by 7 would result in a sum that is divisible by 7. This scenario demonstrates that the sum of 'a' and 'b' would be divisible by 7.

However, if 'a' has a remainder of 1 when divided by 7, and 'b' has a remainder of 2 when divided by 7, the sum of 'a' and 'b' would not be divisible by 7. In this case, the statements together do not provide enough information to conclude whether the sum of 'a' and 'b' is divisible by 7.

Therefore, we can conclude that BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient. The correct answer is (C).

Let's examine each statement individually:

Statement (1): a = 8p, where p is a positive integer.
This statement tells us that a is divisible by 8. However, it does not provide any information about b or its divisibility by 8. Since we need information about both a and b to determine their HCF, statement (1) alone is not sufficient.

Statement (2): b = 7q, where q is a positive integer.
This statement tells us that b is divisible by 7. However, it does not provide any information about a or its divisibility by 7. Similar to statement (1), statement (2) alone is not sufficient.

Now let's consider both statements together:

From the original equation, a = 24b + 56, we can rewrite it as a = 8(3b) + 56. This shows that a is divisible by 8.

Additionally, using statement (2), we have b = 7q, which indicates that b is divisible by 7.

Combining both pieces of information, we know that a is divisible by 8 and b is divisible by 7. However, this is not enough to determine the HCF for a and b. For example, a could be 56 (divisible by 8) and b could be 7 (divisible by 7), resulting in an HCF of 1. Alternatively, a could be 112 (divisible by 8) and b could be 14 (divisible by 7), which would give an HCF of 2.

Since we cannot definitively determine the HCF for a and b based on the given information, the correct answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

To determine the number of multiples of 9 in the sequence S, let's analyze the information provided in each statement:

Statement (1): There are 15 terms in S.
This statement tells us the total number of terms in the sequence S. However, it does not provide any specific information about the values of the terms or their distribution. Since we don't know the exact values of the terms in S, we cannot determine the number of multiples of 9 in the sequence based solely on this statement. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): The greatest term of S is 126.
This statement gives us information about the largest term in the sequence S. However, it does not provide any information about the other terms or their distribution. Knowing the largest term alone is not enough to determine the number of multiples of 9 in the sequence. For example, S could include multiples of 3 that are not multiples of 9. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements:
By combining the information from both statements, we know that there are 15 terms in the sequence S and the largest term is 126. However, even with this combined information, we still don't have any direct information about the specific values of the terms or their distribution. For example, S could contain multiples of 3 that are not multiples of 9. Therefore, the statements together are not sufficient to answer the question.

Since neither statement alone nor the statements together provide enough information to answer the question, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

To determine the digit at the unit's place of 18^{2a + 5b}, we need to consider the cyclicity of the units digit of the number 18.

The units digit of any power of 18 repeats in cycles of four: 8, 4, 2, 6, 8, 4, 2, 6, and so on.

Let's analyze each statement separately:
Statement (1): a is even, b is a multiple of four.
If a is even, 2a is also even. Since the exponent of 18 is 2a + 5b, it will always be even since it contains an even number (2a) and a multiple of four (5b).

In this case, the units digit of 18^{2a + 5b} will always be 4 because the exponent is always even. Therefore, statement (1) alone is sufficient to answer the question.

Statement (2): b = 12

If b = 12, we know the value of b but not the value of a. Without knowing the value of a, we cannot determine the value of 2a + 5b and, therefore, cannot determine the units digit of 18^{2a + 5b}. Thus, statement (2) alone is not sufficient to answer the question.

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

Statement (1): When N is rounded to the nearest hundred, the result is 50 less than the result when N is rounded to the nearest ten.

This statement implies that the tens digit of N is greater than or equal to 5. For example, if N is rounded to the nearest hundred and the result is 100, then when rounded to the nearest ten, the result must be 150.

However, this statement alone does not provide enough information to determine the units digit of N.

Statement (2): N is divisible by 4.

A number is divisible by 4 if its last two digits form a number that is divisible by 4. For example, if N ends in 24, 28, 32, etc., it is divisible by 4.

Knowing that N is divisible by 4 helps us narrow down the possible values for the units digit. However, it still doesn't give us a unique answer.

Combining both statements:

From statement (1), we know that the tens digit of N is greater than or equal to 5. From statement (2), we know that N is divisible by 4, meaning the last two digits must form a number divisible by 4.

Combining these conditions, we can list the possibilities for the tens and units digits:

• 54 (divisible by 4)
• 64 (divisible by 4)
• 74 (not divisible by 4)
• 84 (divisible by 4)
• 94 (not divisible by 4)

Only the numbers 54, 64, and 84 satisfy both conditions. Therefore, the units digit of N could be 4, 6, or 8.

Since we can't determine the units digit with certainty, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

To determine whether n³ - n is divisible by 4, let's analyze each statement separately.

Statement (1): n = 2k + 1, where k is an integer. By substituting this expression for n in n³ - n, we get: (2k + 1)³ - (2k + 1)

Expanding the cube and simplifying, we have: 8k³ + 12k² + 6k + 1 - 2k - 1 8k³ + 12k² + 4k

Factoring out 4 from the expression, we get: 4(2k³ + 3k² + k)

Since 2k³ + 3k² + k is an integer, n³ - n is divisible by 4. Therefore, statement (1) alone is sufficient.

Statement (2): n² + n is divisible by 6. If n² + n is divisible by 6, it means it must be divisible by both 2 and 3.

Let's test some values to see if statement (2) is always true. For n = 1: n² + n = 1² + 1 = 2, which is not divisible by 6. For n = 2: n² + n = 2² + 2 = 6, which is divisible by 6. For n = 3: n² + n = 3² + 3 = 12, which is divisible by 6.

Since statement (2) is not always true, it is not sufficient to answer the question.

Combining both statements, we know that n = 2k + 1, and n² + n is divisible by 6. However, statement (2) is not sufficient on its own, so the information provided is still not enough to determine whether n³ - n is divisible by 4. Thus, the correct answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

To analyze the given statements and determine whether both x and y are divisible by d, let's examine each statement individually:

Statement (1): x + y is divisible by d.
This means that the sum of x and y is divisible by d.
However, we don't have any information about the individual values of x and y or their divisibility by d.
For example, if x = 3, y = 2, and d = 5, the sum x + y is divisible by d, but neither x nor y is individually divisible by d.
Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): x - y is divisible by d.
Similarly, this statement tells us that the difference between x and y is divisible by d.
However, just like in statement (1), we lack information about the divisibility of x and y individually.
For instance, if x = 7, y = 4, and d = 3, the difference x - y is divisible by d, but neither x nor y is divisible by d.
Hence, statement (2) alone is not sufficient to answer the question.

Combining both statements, we have the information that both the sum (x + y) and the difference (x - y) are divisible by d.
This implies that both x and y must be divisible by d.
For example, if x = 6, y = 3, and d = 3, both (x + y) and (x - y) are divisible by d, and x and y are individually divisible by d.
Thus, together, statements (1) and (2) are sufficient to answer the question.

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.

To determine whether the unit digit of the product x * y is greater than 5, we need to consider both statements together, as neither statement alone is sufficient to answer the question.

Let's examine each statement individually:

Statement (1): x is a factor of 85. The prime factorization of 85 is 5 * 17. Since the statement doesn't provide any information about the value of y, we cannot determine the unit digit of the product x * y. For example, if y is 1, the unit digit would be 5, but if y is 9, the unit digit would be 3. Therefore, statement (1) alone is not sufficient.

Statement (2): The units digit of y² and y⁷ is the same. This statement alone doesn't provide any information about the units digit of x or its relation to y. It only focuses on the units digit of y² and y⁷. Without knowing the values of x and y, we cannot determine the unit digit of x * y. Statement (2) alone is also not sufficient.

Combining the statements, we still don't have enough information to determine the unit digit of x * y. The statements provide information about x and y independently, but they don't give us any direct relationship between them. We don't know the values of x or y, their units digits, or any other specific information required to solve the problem.

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

Statement (1): x and y are prime numbers whose sum is a factor of 57.

From this statement, we know that x and y are prime numbers, and their sum is a factor of 57. Since the factors of 57 are 1, 3, 19, and 57, the possible values for the sum of x and y are 1, 3, 19, or 57.

Case 1: If x + y = 1, it means both x and y are 1, but since x < y, this case is not possible.

Case 2: If x + y = 3, then the possible values for x and y are (1, 2). In this case, x is a factor of any even number z because any even number is divisible by 1.

Case 3: If x + y = 19, there are several possible pairs of prime numbers that satisfy this condition, such as (5, 14), (7, 12), (11, 8), and so on. In these cases, x may or may not be a factor of z, depending on the specific values of x, y, and z.

Case 4: If x + y = 57, the only possible pair of prime numbers that satisfies this condition is (19, 38). Again, x may or may not be a factor of z, depending on the specific values of x, y, and z.

Therefore, statement (1) alone is not sufficient to determine whether x is a factor of z.

Statement (2): y is not a factor of z.

From this statement, we know that y is not a factor of z. This does not provide any direct information about x being a factor of z. It is possible that x is a factor of z, or it is not. Therefore, statement (2) alone is not sufficient to determine whether x is a factor of z.

When we consider both statements together, we still do not have enough information to determine whether x is a factor of z. Statement (1) gives us information about the possible values of x and y, but it does not guarantee that x is a factor of z. Statement (2) also does not provide any information about x being a factor of z. Therefore, when considered together, the statements are not sufficient to answer the question.

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