Class 8 Maths Chapter 5 – Number Play

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Practise Class 8 Maths Chapter 5 Number Play MCQs Set 3 based on consecutive numbers, divisibility properties, remainders, and mathematical reasoning with detailed explanations and answers.

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Chapter Information

• Class: 8
• Subject: Mathematics
• Chapter: 5 – Number Play
• Practice Set: 3
• Board: CBSE
• Textbook Reference: NCERT Ganita Prakash, Part I
• Question Type: Multiple-Choice Questions (MCQs)

Introduction

Number Play helps students discover mathematical patterns and understand how numbers behave under different operations. By studying consecutive numbers, divisibility, and remainders, students can develop logical reasoning and solve problems more efficiently.

This practice set focuses on divisibility properties, sums of consecutive numbers, and remainders. Each question includes a detailed explanation to help students understand the mathematical reasoning behind the correct answer.

What Will You Learn?

After completing this practice set, students will be able to:

• Apply divisibility properties to sums and differences.
• Identify patterns in consecutive numbers.
• Calculate remainders without performing complete division.
• Understand the relationship between multiples and divisors.
• Use algebra to explain mathematical statements.

Why Is This Topic Important?

Understanding number properties helps students solve problems without relying entirely on long calculations. These skills are useful for mathematical puzzles, divisibility questions, and developing the algebraic reasoning needed in higher classes.

Exam Relevance

Questions from this chapter may test students’ ability to:

• Determine whether a number is divisible by another number.
• Apply divisibility properties to sums and differences.
• Find remainders using known information.
• Identify patterns involving consecutive integers.
• Explain mathematical statements using examples and algebra.

MCQs with Detailed Solutions

Question 1: Divisibility of a Sum

A number is divisible by 7, and another number is also divisible by 7. Which statement about their sum is always true?

A. Their sum is divisible by 14.

B. Their sum is divisible by 7.

C. Their sum is always odd.

D. Their sum is divisible by 3.

Answer: B. Their sum is divisible by 7.

Concept: The sum of two multiples of the same number is a multiple of that number.

Explanation:

Let the two numbers be (7a) and (7b), where (a) and (b) are integers.

Their sum is:

$$7a+7b=7(a+b)$$

Since (a+b) is an integer, the sum is divisible by 7.

For example:

$$21+35=56$$

$$56\div7=8$$

The sum is divisible by 7, but it is not necessarily divisible by 14, as shown by the example (7+14=21).

Correct Answer: B. Their sum is divisible by 7.

Question 2: Consecutive Numbers

The sum of four consecutive integers is 50. What are the four integers?

A. 10, 11, 12, 13

B. 11, 12, 13, 14

C. 12, 13, 14, 15

D. 9, 10, 11, 12

Answer: B. 11, 12, 13, 14

Concept: Finding consecutive integers using their sum.

Explanation:

Let the four consecutive integers be:

$$n,\ n+1,\ n+2,\ n+3$$

Their sum is 50:

$$n+(n+1)+(n+2)+(n+3)=50$$

Simplifying:

$$4n+6=50$$

Subtract 6 from both sides:

$$4n=44$$

Divide both sides by 4:

$$n=11$$

Therefore, the four integers are:

$$11,\ 12,\ 13,\ 14$$

Verification:

$$11+12+13+14=50$$

Correct Answer: B. 11, 12, 13, 14

Question 3: Finding a Remainder

When a number is divided by 7, the remainder is 3. What will be the remainder when twice that number is divided by 7?

A. 1

B. 3

C. 5

D. 6

Answer: C. 6

Concept: Remainders when a number is multiplied.

Explanation:

Let the number be (N).

Since the remainder is 3 when (N) is divided by 7, we can write:

$$N=7k+3$$

where (k) is an integer.

Twice the number is:

$$2N=2(7k+3)$$

Expanding:

$$2N=14k+6$$

Rewrite this as:

$$2N=7(2k)+6$$

Since 6 is less than 7, the remainder is 6.

For example, if the number is 10:

$$10\div7$$

The remainder is 3.

Twice the number is 20:

$$20\div7=2\text{ remainder }6$$

Correct Answer: C. 6

Question 4: Divisibility by 3 and 9

A number is divisible by 9. Which statement must be true?

A. The number is divisible by 2.

B. The number is divisible by 6.

C. The number is divisible by 3.

D. The number is divisible by 5.

Answer: C. The number is divisible by 3.

Concept: Relationship between divisibility by 3 and 9.

Explanation:

Since:

$$9=3\times3$$

every multiple of 9 is also a multiple of 3.

If a number (N) is divisible by 9, then:

$$N=9k$$

for some integer (k).

Therefore:

$$N=3(3k)$$

This shows that (N) is divisible by 3.

However, it need not be divisible by 2, 6, or 5. For example, 27 is divisible by 9 and 3, but it is not divisible by 2, 6, or 5.

Correct Answer: C. The number is divisible by 3.

Question 5: Divisibility of a Difference

Two numbers are both divisible by 8. Which statement about their difference is always true?

A. Their difference is divisible by 8.

B. Their difference is always positive.

C. Their difference is divisible by 16.

D. Their difference is always odd.

Answer: A. Their difference is divisible by 8.

Concept: The difference of two multiples of the same number.

Explanation:

Let the two numbers be (8a) and (8b), where (a) and (b) are integers.

Their difference is:

$$8a-8b=8(a-b)$$

Since (a-b) is an integer, the difference is divisible by 8.

For example:

$$56-24=32$$

And:

$$32\div8=4$$

The difference is divisible by 8.

The difference does not have to be positive, because the order of subtraction matters. It is also not necessarily divisible by 16 or always odd.

Correct Answer: A. Their difference is divisible by 8.

Important Formulas & Concepts

Divisibility of a Sum

If two numbers are divisible by (k), their sum is also divisible by (k).

If:

$$A=ka,\qquad B=kb$$

then:

$$A+B=k(a+b)$$

Divisibility of a Difference

If two numbers are divisible by (k), their difference is also divisible by (k).

$$A-B=k(a-b)$$

The difference may be positive, negative, or zero.

Four Consecutive Integers

Four consecutive integers can be represented as:

$$n,\ n+1,\ n+2,\ n+3$$

Their sum is:

$$4n+6$$

Remainder Representation

If a number (N) leaves a remainder (r) when divided by (d), then:

$$N=dq+r$$

where (q) is the quotient and (0\leq r<d).

Divisibility by 9 and 3

Every number divisible by 9 is also divisible by 3 because 9 is a multiple of 3.

However, every number divisible by 3 is not necessarily divisible by 9.

Related Practice Questions

Try solving these questions independently.

1. If two numbers are divisible by 12, what can you conclude about their sum?
2. Find four consecutive integers whose sum is 74.
3. A number leaves remainder 4 when divided by 9. What remainder will twice that number leave when divided by 9?
4. Is every number divisible by 6 also divisible by 3? Explain your answer.
5. Two multiples of 11 are subtracted. What can you conclude about their difference?

Mini Quiz Challenge

Question 1

Which number is divisible by both 3 and 5?

A. 124

B. 135

C. 142

D. 221

Answer: B. 135

Explanation:

135 ends in 5, so it is divisible by 5.

The sum of its digits is:

$$1+3+5=9$$

Since 9 is divisible by 3, 135 is also divisible by 3.

Therefore, 135 is divisible by both 3 and 5.

Question 2

What is the sum of four consecutive integers beginning with 16?

A. 66

B. 68

C. 70

D. 72

Answer: C. 70

Explanation:

The four integers are:

$$16,\ 17,\ 18,\ 19$$

Their sum is:

$$16+17+18+19=70$$

Question 3

A number leaves remainder 2 when divided by 5. What remainder will three times that number leave when divided by 5?

A. 1

B. 2

C. 3

D. 4

Answer: A. 1

Explanation:

Let the number be:

$$N=5k+2$$

Three times the number is:

$$3N=3(5k+2)$$

$$3N=15k+6$$

Rewrite the expression:

$$3N=5(3k+1)+1$$

Therefore, the remainder is 1.

Exam Tips

• Use algebra to explain why divisibility properties always work.
• Remember that divisibility is preserved when adding or subtracting multiples of the same number.
• When finding consecutive integers, represent them using a variable.
• In remainder questions, write the number as divisor × quotient + remainder.
• Check whether the remainder is smaller than the divisor.
• Verify every MCQ option before choosing the answer.

Quick Revision Notes

• The sum of two multiples of the same number is divisible by that number.
• The difference of two multiples of the same number is also divisible by that number.
• Four consecutive integers can be represented by (n,n+1,n+2,n+3).
• A remainder must be non-negative and smaller than the divisor.
• Every multiple of 9 is a multiple of 3.
• Algebra helps prove number properties for all integers.

Common Mistakes Students Make

• Assuming that the sum of multiples of a number must be divisible by twice that number.
• Forgetting to include all consecutive integers.
• Confusing the quotient with the remainder.
• Forgetting that a difference can be negative or zero.
• Assuming that divisibility by 3 guarantees divisibility by 9.
• Making arithmetic errors when simplifying algebraic expressions.

Key Takeaways

Divisibility properties help students understand how numbers behave when added or subtracted. Consecutive-number problems can be solved systematically using algebra, while remainder problems become easier when numbers are expressed as a multiple of the divisor plus a remainder.

These techniques strengthen mathematical reasoning and provide a foundation for more advanced number puzzles.

FAQs

1. What happens when two multiples of the same number are added?

Their sum is also divisible by that number.

2. Is the difference of two multiples of 8 always divisible by 8?

Yes. Their difference is another multiple of 8, including when the difference is zero or negative.

3. How can four consecutive integers be represented algebraically?

They can be represented as (n,n+1,n+2,n+3).

4. If a number is divisible by 9, is it also divisible by 3?

Yes, because 9 is divisible by 3.

5. How can we solve remainder problems without long division?

Express the number as the divisor multiplied by an integer, plus its remainder. Then simplify the expression for the required operation.

Conclusion

This practice set covers divisibility of sums and differences, consecutive integers, and remainder calculations from Class 8 Mathematics. The detailed solutions explain how algebra can be used to justify number properties and solve problems accurately.

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