Factors & Multiples for Competitive Exam Success

Factors & Multiples

Q. In a certain number system, the number 12 is represented as 'A' and the number 18 as 'B'. If 'A' is a factor of 'B', which of the following statements is true?

A. A is greater than B
B. B is a multiple of A
C. A and B are equal
D. A is a multiple of B

Solution

'B' (18) is a multiple of 'A' (12) since 18 can be expressed as 12 multiplied by 1.5.

Correct Answer: B — B is a multiple of A

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Q. In a certain number system, the number 12 is represented as 'AB'. If 'A' is a factor of 12 and 'B' is a multiple of 3, which of the following could be the value of 'AB'?

A. 24
B. 36
C. 48
D. 60

Solution

'A' can be 1, 2, 3, 4, 6, or 12 (factors of 12) and 'B' can be 3, 6, 9, 12, etc. The only combination that fits is A=3 and B=12, which gives us 36.

Correct Answer: B — 36

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Q. In a certain number system, the number 12 is represented as 'AB'. If 'A' is a factor of 12 and 'B' is a multiple of 3, which of the following pairs (A, B) is valid?

A. (1, 3)
B. (2, 6)
C. (3, 9)
D. (4, 12)

Solution

In this case, A must be a factor of 12 (1, 2, 3, 4, 6, 12) and B must be a multiple of 3 (3, 6, 9, 12). The pair (2, 6) satisfies both conditions.

Correct Answer: B — (2, 6)

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Q. In a certain number system, the number 12 is represented as 'AB'. If 'A' is a factor of 12 and 'B' is a multiple of 3, which of the following could be the representation of 12?

A. 24
B. 36
C. 48
D. 60

Solution

'A' can be 3 or 4 (factors of 12), and 'B' can be 3, 6, or 9 (multiples of 3). The only combination that fits is 3 and 4, which gives us 24.

Correct Answer: A — 24

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Q. In a certain number, if 12 is added to it, the result is a multiple of 5. What can be inferred about the original number?

A. It is a multiple of 5.
B. It is a multiple of 12.
C. It is one less than a multiple of 5.
D. It is one more than a multiple of 5.

Solution

If the original number is x, then x + 12 is a multiple of 5. This implies that x must be one more than a multiple of 5.

Correct Answer: D — It is one more than a multiple of 5.

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Q. In a certain number, if 12 is added to it, the result is a multiple of 5. Which of the following could be the original number?

A. 3
B. 8
C. 13
D. 17

Solution

If we add 12 to 8, we get 20, which is a multiple of 5.

Correct Answer: B — 8

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Q. In a certain number, if the sum of its digits is a multiple of 3, what can be inferred about the number?

A. It is a prime number.
B. It is an even number.
C. It is a multiple of 3.
D. It is a perfect square.

Solution

A number is divisible by 3 if the sum of its digits is divisible by 3.

Correct Answer: C — It is a multiple of 3.

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Q. In a classroom, if every student has either 2 or 3 pencils, and the total number of pencils is 30, which of the following could be the number of students with 2 pencils?

A. 10
B. 5
C. 15
D. 20

Solution

If there are 10 students with 2 pencils, then there are 10 students with 3 pencils, totaling 30 pencils.

Correct Answer: A — 10

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Q. In a classroom, if the number of students is a multiple of both 4 and 6, which of the following could be the number of students?

A. 20
B. 24
C. 30
D. 36

Solution

The least common multiple of 4 and 6 is 12. Therefore, any multiple of 12 is valid, and 24 is a multiple of 12.

Correct Answer: B — 24

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Q. In a classroom, the teacher has 24 pencils and wants to distribute them equally among students. If each student receives a multiple of 3 pencils, how many students can receive pencils?

A. 6
B. 8
C. 4
D. 3

Solution

The multiples of 3 that can divide 24 are 3, 6, 9, and 12. The maximum number of students that can receive pencils is 8 (3 pencils each).

Correct Answer: B — 8

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Q. In a classroom, the teacher has 48 pencils and wants to distribute them equally among students. If each student receives a multiple of 4 pencils, what is the maximum number of students that can receive pencils?

A. 12
B. 16
C. 8
D. 6

Solution

The maximum number of students is 12, as 48 ÷ 4 = 12.

Correct Answer: B — 16

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Q. In a classroom, the teacher wants to arrange chairs in rows such that each row has the same number of chairs. If there are 48 chairs and the number of rows must be a factor of 48, which of the following is NOT a possible number of rows?

A. 4
B. 6
C. 8
D. 10

Solution

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. 10 is not a factor of 48.

Correct Answer: D — 10

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Q. In a classroom, the teacher wants to arrange chairs in rows such that each row has the same number of chairs. If there are 36 chairs, which of the following is NOT a possible number of chairs per row?

A. 1
B. 2
C. 3
D. 5

Solution

5 is not a factor of 36, hence it cannot be a possible number of chairs per row.

Correct Answer: D — 5

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Q. In a classroom, the teacher wants to arrange students in groups such that each group has the same number of students. If there are 36 students, which of the following is NOT a possible group size? (2023)

A. 1
B. 2
C. 3
D. 10

Solution

The number 10 is not a factor of 36, hence it cannot be a group size.

Correct Answer: D — 10

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Q. In a classroom, the teacher wants to arrange students in rows such that each row has the same number of students. If there are 24 students, which of the following arrangements is NOT possible?

A. 6 rows of 4 students
B. 8 rows of 3 students
C. 12 rows of 2 students
D. 5 rows of 5 students

Solution

5 rows of 5 students would require 25 students, which is not possible with only 24 students.

Correct Answer: D — 5 rows of 5 students

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Q. In a sequence of numbers where each number is a multiple of 3, which of the following cannot be a member of this sequence?

A. 9
B. 15
C. 22
D. 27

Solution

22 is not a multiple of 3, while 9, 15, and 27 are.

Correct Answer: C — 22

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Q. In a sequence of numbers where each number is a multiple of 7, which of the following could be the 5th number in the sequence?

A. 28
B. 35
C. 42
D. 49

Solution

The 5th number in the sequence of multiples of 7 is 7 * 5 = 35, so 42 is the next multiple after 35.

Correct Answer: C — 42

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Q. In a sequence of numbers where each number is a multiple of 8, which of the following could be the 5th number in the sequence?

A. 32
B. 40
C. 56
D. 72

Solution

The 5th number in a sequence of multiples of 8 could be 32 (8 x 4), while the others are not multiples of 8.

Correct Answer: A — 32

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Q. In a sequence of numbers where each number is a multiple of 9, which of the following numbers cannot be in the sequence?

A. 27
B. 45
C. 81
D. 50

Solution

50 is not a multiple of 9, while 27, 45, and 81 are all multiples of 9.

Correct Answer: D — 50

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Q. In a set of numbers, if the number 30 is a multiple of a certain number 'X', which of the following could be 'X'?

A. 5
B. 7
C. 10
D. 12

Solution

Both 5 and 10 are factors of 30, making them valid options for 'X'.

Correct Answer: A — 5

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Q. In the context of factors and multiples, which of the following statements is true?

A. Every multiple of a number is also a factor of that number.
B. A factor of a number is always greater than the number itself.
C. The least common multiple of two numbers is always greater than or equal to both numbers.
D. Factors of a number can only be positive.

Solution

The least common multiple (LCM) of two numbers is defined as the smallest number that is a multiple of both. Therefore, it is always greater than or equal to both numbers.

Correct Answer: C — The least common multiple of two numbers is always greater than or equal to both numbers.

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Q. What is the GCF of 24 and 36?

A. 6
B. 12
C. 18
D. 24

Solution

The GCF of 24 and 36 is 12, as it is the largest number that divides both without leaving a remainder.

Correct Answer: B — 12

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Q. What is the least common multiple (LCM) of 4 and 5?

A. 10
B. 15
C. 20
D. 25

Solution

The LCM of 4 and 5 is 20, as it is the smallest number that both 4 and 5 divide evenly into.

Correct Answer: C — 20

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Q. What is the smallest multiple of 9 that is greater than 50?

A. 54
B. 63
C. 72
D. 81

Solution

The smallest multiple of 9 greater than 50 is 54.

Correct Answer: A — 54

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Q. Which of the following is a necessary condition for a number to be a multiple of 18?

A. It must be even.
B. It must be a multiple of 9.
C. It must be a multiple of 3.
D. It must be a multiple of 6.

Solution

Since 18 = 2 × 9, being a multiple of 9 is necessary for a number to be a multiple of 18.

Correct Answer: B — It must be a multiple of 9.

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Q. Which of the following is a true statement about prime numbers?

A. Every prime number is a factor of 1.
B. Every prime number is a multiple of itself.
C. All prime numbers are even.
D. Prime numbers have exactly two distinct positive divisors.

Solution

Prime numbers are defined as having exactly two distinct positive divisors: 1 and the number itself.

Correct Answer: D — Prime numbers have exactly two distinct positive divisors.

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Q. Which of the following is a true statement regarding multiples?

A. All multiples of a number are even.
B. Multiples of a number can be negative.
C. The first multiple of any number is the number itself.
D. Multiples of a number are always prime.

Solution

The first multiple of any number is indeed the number itself, while the other statements are false.

Correct Answer: C — The first multiple of any number is the number itself.

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Q. Which of the following is NOT a characteristic of multiples?

A. They are always greater than or equal to the original number.
B. They can be negative.
C. They are always odd.
D. They can be zero.

Solution

Multiples can be both odd and even; hence, they are not always odd.

Correct Answer: C — They are always odd.

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Q. Which of the following is NOT a factor of 60?

A. 2
B. 3
C. 5
D. 7

Solution

The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. 7 is not among them.

Correct Answer: D — 7

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Q. Which of the following is the best definition of a factor?

A. A number that can divide another number without leaving a remainder.
B. A number that is a multiple of another number.
C. A number that is less than another number.
D. A number that is greater than one.

Solution

A factor is defined as a number that divides another number without leaving a remainder.

Correct Answer: A — A number that can divide another number without leaving a remainder.

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Factors & Multiples

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