Test: Divisibility/Multiples/Factors - GMAT MCQ

(1) X is a multiple of 42.

Prime factors of 42 are 2, 3 and 7. If x is 42, unique prime factors are less than 4, if x is 462, unique prime factors are 2,3,5,7 and 11 i.e. greater than 4. Therefore, not sufficient.

(2) X is a multiple of 98.

Prime factors of 98 are 2 and 7. A multiple of 98 can have more than or less than 4 unique prime factors. Not sufficient.

Merging them, we get that x is a multiple of both, 98 and 42. The prime factors of LCM are 2, 3 and 7 i.e. less than 4 but if we multiple the LCM itself by 55, then we would have more than 4 unique prime factors. Therefore, not sufficient.

E is the correct answer.

Statement 1

(1) n is a multiple of 6

We cannot conclude whether x is a multiple of 12.

Ex. x = 6; Is 6 a multiple of 12 -- No
x = 12; Is 12 a multiple of 12 -- Yes

The statement is not sufficient and we can eliminate A and D.

Statement 2

(2) n is a multiple of 24

If n is a multiple of 24, the number is also a multiple of 12.

The statement is sufficient.

(1) a = 2

If we substitute a = 2 into the expression 74a + b, we have 74(2) + b = 148 + b. Without knowing the value of b, we cannot determine the remainder when dividing by 4. Therefore, statement (1) alone is not sufficient to answer the question.

(2) b = 3

If we substitute b = 3 into the expression 74a + b, we have 74a + 3. However, we don't have any information about the value of a, and without knowing the value of a, we cannot determine the remainder when dividing by 4. Therefore, statement (2) alone is not sufficient to answer the question.

When considering each statement individually, neither statement alone is sufficient to determine the remainder when 74a + b is divided by 4.

Let's consider both statements together:

From statement (1), we know that a = 2. From statement (2), we know that b = 3.

Substituting these values into the expression 74a + b, we have 74(2) + 3 = 148 + 3 = 151.

To find the remainder when 151 is divided by 4, we divide 151 by 4:

151 ÷ 4 = 37 remainder 3.

Therefore, both statements together are sufficient to determine the remainder when 74a + b is divided by 4.

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

Statement (1): x+z is divisible by 5.
This statement alone does not provide any information about whether x+y is divisible by 5. We don't have any direct relationship between x+z and x+y.

Statement (2): y+z is divisible by 5.
Similarly, this statement alone does not give us any information about the divisibility of x+y by 5. There is no direct relationship between y+z and x+y.

Since neither statement alone is sufficient to answer the question, and the statements do not provide enough information when considered together, the answer is (E): Statements (1) and (2) together are not sufficient to answer the question asked, and additional data are needed.

(1) The three-digit number whose hundreds digit is (x+2), tens digit is (y+1), and units digit is z is divisible by 9.

If this three-digit number is divisible by 9, it means that the sum of its digits is divisible by 9. From the given information, we have (x+2) + (y+1) + z. However, we don't have any specific information about the values of x, y, and z, so we cannot determine if their sum is divisible by 9. Therefore, statement (1) alone is not sufficient to answer the question.

(2) xyz is divisible by 7.

This statement tells us that the three-digit number xyz is divisible by 7. However, divisibility by 7 does not provide any information about divisibility by 9. Therefore, statement (2) alone is not sufficient to answer the question.

When considering each statement individually, neither statement alone is sufficient to determine the remainder when xyz is divided by 9.

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

(1) p - 5 is a multiple of 3.

From this statement, we can deduce that p is 5 more than a multiple of 3. However, this information alone does not give us any specific information about the tens digit of p or its divisibility by 3.

(2) p - 11 is a multiple of 3.

From this statement, we can deduce that p is 11 more than a multiple of 3. Again, this information alone does not provide us with any specific information about the tens digit of p or its divisibility by 3.

When considering both statements together, we still don't have enough information to determine the divisibility of the tens digit of p by 3. The statements only provide information about the difference between p and a multiple of 3, but not about the specific value or range of p.

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

(1) x, y, z are consecutive.

If x, y, and z are consecutive, it means they are consecutive positive integers. In this case, we have two possible scenarios:

• If one of x, y, or z is divisible by 2 and another one is divisible by 3, then xyz will be divisible by 6 because it will have both prime factors 2 and 3.
• If none of x, y, or z is divisible by 2 or 3, then xyz will not be divisible by 6.

Therefore, statement (1) alone is sufficient to determine if xyz is divisible by 6.

(2) x + y + z is a multiple of 3.

If the sum of x, y, and z is a multiple of 3, it implies that at least one of x, y, or z is divisible by 3. However, this information alone does not tell us anything about whether xyz is divisible by 2. There could be cases where x, y, and z are not consecutive but still result in xyz being divisible by 6.

When considering each statement individually, we find that statement (1) alone is sufficient to determine if xyz is divisible by 6, while statement (2) alone is not sufficient.

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

(1) 14 is a factor of x.

If 14 is a factor of x, it means that x is divisible by 14. Since 14 is a factor of x, and 4 is a factor of x, we can conclude that x must be a multiple of 4 and 14. Since 42 is a multiple of both 4 and 14, if x is divisible by both 4 and 14, then xy will be divisible by 42. Therefore, statement (1) alone is sufficient to answer the question.

(2) 25 is a factor of y.

If 25 is a factor of y, it means that y is divisible by 25. However, this information alone does not tell us anything about x or whether xy is divisible by 42. Therefore, statement (2) alone is not sufficient to answer the question.

When considering each statement individually, we find that statement (1) alone is sufficient to determine if 42 is a factor of xy, while statement (2) alone is not sufficient.

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

Let's analyze the given statements:

Statement (1): x is a multiple of 9. This statement tells us that x is a multiple of 3, which is one of the prime factors of 105. Therefore, if statement (1) is true, xy will be a multiple of 105 regardless of the value of y.

Statement (2): y is a multiple of 25. This statement tells us that y is a multiple of 5, another prime factor of 105. However, it does not provide any information about whether x is a multiple of 3 or 7, so we cannot determine if xy is a multiple of 105 based on statement (2) alone.

Combining both statements, we know that x is a multiple of 3 and y is a multiple of 5. However, we still don't know if x is a multiple of 7. Without knowing if x is divisible by 7, we cannot conclusively determine if xy is a multiple of 105.

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

(1) Each of the numbers 3 and 8 is a divisor of s.

If both 3 and 8 are divisors of s, it means that s is divisible by both 3 and 8. Since 24 is a multiple of both 3 and 8, if s is divisible by both 3 and 8, then it is also divisible by 24. Therefore, statement (1) alone is sufficient to answer the question.

(2) Each of the numbers 4 and 6 is a divisor of s.

If both 4 and 6 are divisors of s, it means that s is divisible by both 4 and 6. However, this information alone does not tell us if s is divisible by 24. For example, s could be 12, which is divisible by 4 and 6 but not by 24. Therefore, statement (2) alone is not sufficient to answer the question.

When considering each statement individually, we find that statement (1) alone is sufficient to determine if 24 is a divisor of s, while statement (2) alone is not sufficient.

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