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Practise Class 8 Maths Chapter 5 Number Play MCQs Set 4 with questions on digit patterns, reversing numbers, divisibility by 11, number properties, and mathematical reasoning with detailed solutions.
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Chapter Information
- Class: 8
- Subject: Mathematics
- Chapter: 5 – Number Play
- Practice Set: 4
- Board: CBSE
- Textbook Reference: NCERT Ganita Prakash, Part I
- Question Type: Multiple-Choice Questions (MCQs)
Introduction
Numbers contain interesting patterns that can be discovered by examining their digits and mathematical properties. Reversing digits, testing divisibility, and exploring number relationships help students develop logical thinking and recognise mathematical patterns.
This practice set contains five original questions involving reversed numbers, digit sums, divisibility by 11, and properties of squares. Each question includes a detailed explanation to help students understand the reasoning behind the correct answer.
What Will You Learn?
After completing this practice set, students will be able to:
- Understand the effect of reversing the digits of a two-digit number.
- Apply digit-sum rules to check divisibility.
- Test divisibility by 11.
- Identify patterns in perfect squares.
- Use mathematical reasoning to solve number puzzles.
Why Is This Topic Important?
Number patterns help students discover relationships that are not always obvious from ordinary calculations. Learning to analyse digits and divisibility makes calculations easier and builds a foundation for algebra and more advanced mathematical reasoning.
Exam Relevance
Questions from number-based mathematical activities may involve:
- Reversing the digits of numbers.
- Applying divisibility rules.
- Identifying properties of perfect squares.
- Analysing numerical patterns.
- Explaining why a mathematical statement is always true.
MCQs with Detailed Solutions
Question 1: Reversing the Digits of a Number
A two-digit number has 4 in the tens place and 7 in the ones place. What is the difference between the reversed number and the original number?
A. 9
B. 18
C. 27
D. 36
Answer: C. 27
Concept: Place value and reversing digits.
Explanation:
The original number is:
$$10\times4+7=47$$
After reversing the digits, the number becomes:
$$10\times7+4=74$$
The difference is:
$$74-47=27$$
Therefore, the difference between the reversed number and the original number is 27.
Correct Answer: C. 27
Question 2: Divisibility Using the Sum of Digits
Which of the following numbers is divisible by 9?
A. 5,832
B. 6,214
C. 7,325
D. 8,146
Answer: A. 5,832
Concept: Divisibility rule for 9.
Explanation:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Option A: 5,832
$$5+8+3+2=18$$
Since 18 is divisible by 9, 5,832 is divisible by 9.
Verification:
$$5,832\div9=648$$
Option B: 6,214
$$6+2+1+4=13$$
13 is not divisible by 9.
Option C: 7,325
$$7+3+2+5=17$$
17 is not divisible by 9.
Option D: 8,146
$$8+1+4+6=19$$
19 is not divisible by 9.
Only 5,832 satisfies the divisibility rule.
Correct Answer: A. 5,832
Question 3: Divisibility by 11
Which of the following numbers is divisible by 11?
A. 352
B. 425
C. 638
D. 742
Answer: A. 352
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 352, the digits in alternate positions give:
$$3+2=5$$
The remaining digit is:
$$5$$
The difference is:
$$5-5=0$$
Since the difference is 0, 352 is divisible by 11.
Verification:
$$352\div11=32$$
Now check the other options.
For 425:
$$4+5-2=7$$
For 638:
$$6+8-3=11$$
For 742:
$$7+2-4=5$$
The differences for 425 and 742 are not multiples of 11. For 638, the difference is 11, so 638 is also divisible by 11.
Therefore, both 352 and 638 are divisible by 11. The original question has two correct options.
Corrected Question: Which of the following numbers is divisible by 11?
A. 352
B. 425
C. 637
D. 742
For option C:
$$6+7-3=10$$
Only option A has a difference of 0, while the other options have differences of 7, 10, and 5, respectively.
Correct Answer: A. 352
Question 4: Identifying a Perfect Square
Which of the following numbers is a perfect square?
A. 121
B. 135
C. 150
D. 175
Answer: A. 121
Concept: Perfect squares.
Explanation:
A perfect square is a number obtained by multiplying an integer by itself.
For option A:
$$11\times11=121$$
Therefore:
$$11^2=121$$
The other options are not perfect squares:
- 135 lies between (11^2=121) and (12^2=144).
- 150 lies between (12^2=144) and (13^2=169).
- 175 lies between (13^2=169) and (14^2=196).
Therefore, 121 is the only perfect square among the options.
Correct Answer: A. 121
Question 5: A Property of Reversed Two-Digit Numbers
A two-digit number has tens digit (a) and ones digit (b). Which expression represents the difference between the original number and its reversed form?
A. (9(a-b))
B. (10(a+b))
C. (11(a+b))
D. (9(a+b))
Answer: A. (9(a-b))
Concept: Algebraic representation of two-digit numbers.
Explanation:
The original number is:
$$10a+b$$
The reversed number is:
$$10b+a$$
The difference between the original number and the reversed number is:
$$10a+b-(10b+a)$$
Simplifying:
$$10a+b-10b-a$$
$$=9a-9b$$
Taking out the common factor 9:
$$=9(a-b)$$
Therefore, the difference is represented by (9(a-b)).
This also explains why the difference between a two-digit number and its reversed form is always divisible by 9.
Correct Answer: A. (9(a-b))
Important Formulas & Concepts
Place Value of a Two-Digit Number
If the tens digit is (a) and the ones digit is (b), the number is:
$$10a+b$$
The reversed number is:
$$10b+a$$
Difference Between a Number and Its Reversed Form
Subtracting the reversed number from the original number gives:
$$10a+b-(10b+a)$$
Therefore:
$$9(a-b)$$
Hence, the difference is divisible by 9.
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 its digits in alternate positions is 0 or a multiple of 11.
Perfect Squares
A perfect square is the product of an integer multiplied by itself.
Examples:
$$5^2=25$$
$$9^2=81$$
$$11^2=121$$
Related Practice Questions
Try solving these questions independently.
- Reverse the digits of 63 and find the difference between the two numbers.
- Check whether 7,281 is divisible by 9 using the digit-sum rule.
- Determine whether 814 is divisible by 11.
- Find the square of 14.
- Explain algebraically why the difference between a two-digit number and its reversed form is divisible by 9.
Mini Quiz Challenge
Question 1
What is the reversed form of 86?
A. 68
B. 76
C. 88
D. 96
Answer: A. 68
Explanation:
Reversing the digits of 86 gives 68.
Question 2
Which number is a perfect square?
A. 144
B. 145
C. 155
D. 165
Answer: A. 144
Explanation:
$$12\times12=144$$
Therefore, 144 is a perfect square.
Question 3
Which of the following numbers is divisible by 11?
A. 517
B. 526
C. 634
D. 745
Answer: A. 517
Explanation:
For 517, the difference between the sums of alternate digits is:
$$5+7-1=11$$
Since 11 is a multiple of 11, 517 is divisible by 11.
Verification:
$$517\div11=47$$
Exam Tips
- Identify the place value of each digit before reversing a number.
- Apply divisibility rules carefully and check the arithmetic.
- For two-digit reversed-number problems, use algebra to identify patterns.
- Remember that a perfect square is an integer multiplied by itself.
- Check every answer option to ensure that only one option is correct.
- Verify numerical answers using multiplication or division when possible.
Quick Revision Notes
- A two-digit number with digits (a) and (b) is (10a+b).
- Its reversed form is (10b+a).
- Their difference is (9(a-b)).
- The digit-sum rule helps test divisibility by 9.
- The alternate-digit-sum rule helps test divisibility by 11.
- A perfect square is the square of an integer.
Common Mistakes Students Make
- Confusing the tens digit with the ones digit.
- Reversing a number incorrectly.
- Forgetting the place-value multiplier of 10.
- Applying divisibility rules without checking the digit calculations.
- Confusing a perfect square with any even or odd number.
- Failing to check all four options before selecting an answer.
Key Takeaways
Analysing digits reveals useful mathematical patterns. Reversing a two-digit number produces a difference that is divisible by 9, while divisibility rules make it easier to check numbers without long division. Understanding perfect squares also helps students recognise important numerical relationships.
FAQs
1. How do we represent a two-digit number algebraically?
If its tens digit is (a) and its ones digit is (b), the number is represented by (10a+b).
2. Why is the difference between a two-digit number and its reversed form divisible by 9?
The difference simplifies to (9(a-b)), which is a multiple of 9.
3. How can we check divisibility by 11?
Calculate the difference between the sums of digits in alternate positions. A difference of 0 or a multiple of 11 indicates divisibility.
4. What is a perfect square?
A perfect square is a number obtained by multiplying an integer by itself, such as (12^2=144).
5. Is every number divisible by 9 also divisible by 3?
Yes. Every multiple of 9 is also a multiple of 3.
Conclusion
This practice set explores reversed numbers, divisibility rules, perfect squares, and algebraic number patterns. The detailed explanations help students understand the reasoning behind each answer and develop confidence in solving number-based problems.
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