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Practise Class 8 Maths Chapter 5 Number Play MCQs Set 2 with questions on consecutive numbers, number patterns, parity, divisibility rules, and mathematical reasoning with detailed answers.
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Chapter Information
- Class: 8
- Subject: Mathematics
- Chapter: 5 – Number Play
- Practice Set: 2
- Board: CBSE
- Textbook Reference: NCERT Ganita Prakash, Part I
- Question Type: Multiple-Choice Questions (MCQs)
Introduction
Numbers have many interesting properties that can be understood through patterns and logical reasoning. The chapter Number Play encourages students to explore consecutive numbers, divisibility, even and odd numbers, and the relationships between numbers.
In this practice set, students will investigate sums of consecutive numbers, determine whether expressions are even or odd, and apply divisibility rules. These original practice questions are designed to strengthen conceptual understanding and mathematical reasoning.
What Will You Learn?
After completing this practice set, students will be able to:
- Identify sums of consecutive positive integers.
- Understand patterns involving consecutive numbers.
- Determine the parity of sums of consecutive integers.
- Apply divisibility rules for 9 and 11.
- Use mathematical reasoning to solve number problems.
Why Is This Topic Important?
Number patterns help students discover mathematical rules rather than relying only on calculations. Understanding consecutive numbers and divisibility also helps students solve puzzles, recognise relationships, and explain why a mathematical statement is always true.
Exam Relevance
Questions from this chapter may test students’ ability to:
- Express numbers as sums of consecutive integers.
- Identify even and odd results.
- Apply divisibility tests.
- Recognise number patterns.
- Justify mathematical statements using logical reasoning.
MCQs with Detailed Solutions
Question 1: Sum of Two Consecutive Numbers
Which of the following numbers can be expressed as the sum of two consecutive positive integers?
A. 24
B. 32
C. 37
D. 40
Answer: C. 37
Concept: Sum of two consecutive positive integers.
Explanation:
Let the two consecutive positive integers be (n) and (n+1).
Their sum is:
$$n+(n+1)=2n+1$$
Since (2n) is even, (2n+1) is odd.
Therefore, the sum of two consecutive integers is always odd.
Now check option C:
$$18+19=37$$
Both 18 and 19 are positive consecutive integers. Their sum is 37.
The other options are even numbers, so they cannot be expressed as the sum of two consecutive integers.
Correct Answer: C. 37
Question 2: Parity of Four Consecutive Integers
What can we conclude about the sum of any four consecutive integers?
A. It is always odd.
B. It is always even.
C. It is always divisible by 5.
D. It is always divisible by 8.
Answer: B. It is always even.
Concept: Parity of consecutive integers.
Explanation:
Let four consecutive integers be:
$$n,\ n+1,\ n+2,\ n+3$$
Their sum is:
$$S=n+(n+1)+(n+2)+(n+3)$$
Simplifying:
$$S=4n+6$$
Taking out the factor 2:
$$S=2(2n+3)$$
Since (2n+3) is an integer, the sum is divisible by 2.
Therefore, the sum of any four consecutive integers is always even.
For example:
$$3+4+5+6=18$$
The result is even.
Correct Answer: B. It is always even.
Question 3: Divisibility Rule for 9
Which of the following numbers is divisible by 9?
A. 6,352
B. 4,725
C. 5,416
D. 7,325
Answer: B. 4,725
Concept: Divisibility rule for 9.
Explanation:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Let us check each option.
Option A: 6,352
$$6+3+5+2=16$$
Since 16 is not divisible by 9, 6,352 is not divisible by 9.
Option B: 4,725
$$4+7+2+5=18$$
Since 18 is divisible by 9, 4,725 is divisible by 9.
Verification:
$$4,725\div9=525$$
Option C: 5,416
$$5+4+1+6=16$$
Since 16 is not divisible by 9, 5,416 is not divisible by 9.
Option D: 7,325
$$7+3+2+5=17$$
Since 17 is not divisible by 9, 7,325 is not divisible by 9.
Therefore, only 4,725 is divisible by 9.
Correct Answer: B. 4,725
Question 4: Divisibility Rule for 11
Which of the following numbers is divisible by 11?
A. 2,728
B. 3,415
C. 4,526
D. 5,632
Answer: A. 2,728
Concept: Divisibility rule for 11.
Explanation:
A number is divisible by 11 if the difference between the sums of digits in alternate positions is 0 or a multiple of 11.
For 2,728, the digits in alternate positions give:
$$8+7=15$$
The sum of the remaining digits is:
$$2+2=4$$
The difference is:
$$15-4=11$$
Since 11 is divisible by 11, the number 2,728 is divisible by 11.
Verification:
$$2,728\div11=248$$
The quotient is a whole number.
Correct Answer: A. 2,728
Question 5: Sum of Three Consecutive Integers
Which statement about the sum of three consecutive integers is always true?
A. Their sum is always odd.
B. Their sum is always divisible by 3.
C. Their sum is always divisible by 5.
D. Their sum is always a prime number.
Answer: B. Their sum is always divisible by 3.
Concept: Algebraic representation of consecutive integers.
Explanation:
Let three consecutive integers be:
$$n,\ n+1,\ n+2$$
Their sum is:
$$S=n+(n+1)+(n+2)$$
Simplifying:
$$S=3n+3$$
Taking out the common factor 3:
$$S=3(n+1)$$
Since (n+1) is an integer, the sum is always divisible by 3.
For example:
$$7+8+9=24$$
And:
$$24\div3=8$$
The sum is divisible by 3, so option B is correct.
Correct Answer: B. Their sum is always divisible by 3.
Important Formulas & Concepts
Sum of Two Consecutive Integers
If the integers are (n) and (n+1), then:
$$n+(n+1)=2n+1$$
Their sum is always odd.
Sum of Three Consecutive Integers
For three consecutive integers:
$$n,\ n+1,\ n+2$$
Their sum is:
$$3n+3=3(n+1)$$
Therefore, their sum is always divisible by 3.
Sum of Four Consecutive Integers
For four consecutive integers:
$$n,\ n+1,\ n+2,\ n+3$$
Their sum is:
$$4n+6=2(2n+3)$$
Therefore, their sum is always even.
Divisibility Rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Divisibility Rule for 11
A number is divisible by 11 if the difference between the sums of digits in alternate positions is 0 or a multiple of 11.
Even and Odd Numbers
- An even integer can be expressed as (2n).
- An odd integer can be expressed as (2n+1).
Here, (n) is an integer.
Related Practice Questions
Try solving these questions independently.
- Express 45 as the sum of two consecutive positive integers.
- Is the sum of six consecutive integers always even? Explain your reasoning.
- Check whether 8,127 is divisible by 9 using the digit-sum rule.
- Check whether 6,237 is divisible by 11 using the alternate-digit-sum rule.
- Show algebraically why the sum of five consecutive integers is always divisible by 5.
Mini Quiz Challenge
Question 1
Which of the following is the sum of two consecutive positive integers?
A. 18
B. 25
C. 36
D. 42
Answer: B. 25
Explanation:
$$12+13=25$$
The numbers 12 and 13 are consecutive positive integers.
Question 2
What is the sum of the first three consecutive integers beginning with 14?
A. 42
B. 43
C. 44
D. 45
Answer: C. 45
Explanation:
The three consecutive integers are 14, 15, and 16.
$$14+15+16=45$$
Question 3
Which of the following numbers is divisible by 11?
A. 1,234
B. 2,431
C. 3,146
D. 4,215
Answer: B. 2,431
Explanation:
For 2,431, add the digits in alternate positions:
$$1+4=5$$
Add the remaining digits:
$$3+2=5$$
The difference is:
$$5-5=0$$
Therefore, 2,431 is divisible by 11.
Exam Tips
- Represent consecutive integers using variables before solving.
- Check whether a question asks about two, three, four, or more consecutive integers.
- Use algebra to prove a pattern instead of checking only one example.
- For divisibility by 9, calculate the complete digit sum.
- For divisibility by 11, arrange digits into alternate positions carefully.
- Check every answer option before choosing the correct answer.
Quick Revision Notes
- The sum of two consecutive integers is always odd.
- The sum of three consecutive integers is always divisible by 3.
- The sum of four consecutive integers is always even.
- The sum of five consecutive integers is always divisible by 5.
- The digit-sum rule helps test divisibility by 9.
- The alternate-digit-sum rule helps test divisibility by 11.
Common Mistakes Students Make
- Assuming a pattern is always true after checking just one example.
- Confusing the sum of consecutive integers with their product.
- Forgetting to use algebra when proving a general statement.
- Adding the wrong digits when checking divisibility by 11.
- Selecting an MCQ answer without checking every option.
- Making arithmetic errors while calculating digit sums.
Key Takeaways
Consecutive numbers follow predictable patterns. Algebra helps explain why the sum of three consecutive integers is divisible by 3 and why the sum of four consecutive integers is even. Divisibility rules for 9 and 11 provide useful shortcuts for checking numbers without performing long division.
FAQs
1. What are consecutive integers?
Consecutive integers are integers that follow one another without a gap, such as 8, 9, and 10.
2. Why is the sum of two consecutive integers always odd?
If the integers are (n) and (n+1), their sum is (2n+1), which is odd.
3. Why is the sum of three consecutive integers divisible by 3?
Their sum can be written as (3(n+1)), which is a multiple of 3.
4. How can we check divisibility by 9?
Add all the digits. If the sum is divisible by 9, the original number is divisible by 9.
5. How can we check divisibility by 11?
Find the difference between the sums of digits in alternate positions. If the difference is 0 or a multiple of 11, the number is divisible by 11.
Conclusion
This practice set explores consecutive integers, parity, algebraic patterns, and divisibility rules from Class 8 Mathematics. Solving these original MCQs with detailed explanations helps students understand the reasoning behind number properties and prepare for more challenging problems.
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