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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
Understanding "Factors & Multiples" is crucial for students preparing for various exams in India. This topic forms the foundation for many mathematical concepts and is frequently tested in objective questions. Practicing MCQs related to factors and multiples not only enhances conceptual clarity but also boosts your confidence in tackling important questions during exams.
What You Will Practise Here
Definition and identification of factors and multiples
Prime factors and their significance
Finding the least common multiple (LCM) and greatest common divisor (GCD)
Applications of factors and multiples in problem-solving
Word problems involving factors and multiples
Common patterns and tricks for quick calculations
Practice questions with detailed explanations and answers
Exam Relevance
The topic of factors and multiples is a staple in CBSE and State Board examinations, as well as competitive exams like NEET and JEE. Questions often include finding LCM and GCD, identifying factors of given numbers, and solving real-world problems. Familiarity with this topic can significantly improve your performance, as it frequently appears in both objective and subjective formats.
Common Mistakes Students Make
Confusing factors with multiples, leading to incorrect answers
Overlooking the importance of prime factorization in solving problems
Miscalculating LCM and GCD due to lack of practice
Ignoring word problems that require a clear understanding of the concepts
FAQs
Question: What are factors and multiples? Answer: Factors are numbers that divide another number exactly, while multiples are the result of multiplying a number by an integer.
Question: How can I find the LCM of two numbers? Answer: The LCM can be found using the prime factorization method or by listing the multiples of the numbers until you find the smallest common one.
Start your journey towards mastering factors and multiples today! Solve practice MCQs to test your understanding and improve your exam readiness. Remember, consistent practice is the key to success!
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