Exercise 5.1

NCERT Solutions for Class 8 Maths Chapter 5 Number Play - Exercise 5.1

Mastering the logic behind numbers is simple with our NCERT Solutions for Class 8 Maths Ch 5: Playing with Numbers – Exercise 5.1. In this chapter, we explore numbers in their generalized forms and solve interesting puzzles where letters represent digits in arithmetic operations.

Class 8 Math Lessons (Ch 5): Learn to Solve Number Puzzles with Ease! The solutions to Exercise 5.1 will be presented in a step-by-step approach to give students a clear understanding of how to use place value logic to find missing digits. These solutions are in accordance with the CBSE Guidelines to provide the best path for improving analytical skills. Additionally, these solutions build the foundation for algebraic thinking and mathematical reasoning.

1.0Download Class 8 Maths Chapter 5 Ex 5.1 NCERT Solutions PDF

You can download our simple NCERT Solutions for Class 8 Maths Chapter 5 Exercise 5.1 as a PDF. Ideal for rapid reference and offline study.

NCERT Solutions Class 8 Maths Chapter 5 Ex 5.1

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2.0 Detailed NCERT Class 8 Maths Chapter 5 Solutions of Exercise 5.1

1.The sum of four consecutive numbers is 34. What are these numbers?

Sol. Let $x, x + 1, x + 2$ , and $x + 3$ be the four consecutive numbers, respectively. $∴ x + (x + 1) + (x + 2) + (x + 3) = 34$ $\Rightarrow 4 x + 6 = 34$ $\Rightarrow 4 x = 34 - 6$ $\Rightarrow 4 x = 28$ $\Rightarrow x = 7$ $∴ x + 1 = 7 + 1 = 8$ $x + 2 = 7 + 2 = 9$ and $x + 3 = 7 + 3 = 10$ Thus, the four consecutive numbers are 7, 8, 9, and 10. 2. Suppose $p$ is the greatest of five consecutive numbers. Describe the other four numbers in terms of p. Sol. Given p is the greatest of five consecutive numbers.

The other four numbers in terms of p are $(p - 1), (p - 2), (p - 3)$ , and ( $p - 4$ ). $p - 1$ is the second largest number $p - 2$ is the third largest number p - 3 is the second smallest number $p - 4$ is the smallest number $∴ p > (p - 1) > (p - 2) > (p - 3) > (p - 4)$ . 3. For each statement below, determine whether it is always true, sometimes true, or never true. Explain your answer. Mention examples and non-examples as appropriate. Justify your claim using algebra. (i) The sum of two even numbers is a multiple of 3 . (ii) If a number is not divisible by 18 , then it is also not divisible by 9 . (iii) If two numbers are not divisible by 6 , then their sum is not divisible by 6 . (iv) The sum of a multiple of 6 and a multiple of 9 is a multiple of 3 . (v) The sum of a multiple of 6 and a multiple of 3 is a multiple of 9 .

Sol. (i) Sometimes true, the sum of two even numbers is a multiple of 3 .

Examples: $2 + 4 = 6, 8 + 10 = 18, 14 + 16 = 30$ are multiples of 3 . $2 + 6 = 8, 4 + 10 = 14$ are not multiples of 3 .

(ii) Sometimes true, if a number is not divisible by 18 , then it is also not divisible by 9 .

Examples: 27 is not divisible by 18 , but 27 is divisible by 9 . True

40 is not divisible by 18, also it is not divisible by 9 . False (iii) Never true, if two numbers are not divisible by 6, then their sum is not divisible by 6 .

Examples: 8 and 10 are not divisible by 6 . The sum of two numbers $= 8 + 10 =$ 18 , is divisible by 6 . 10 and 13 are not divisible by 6 . The sum of 10 and $13 = 10 + 13 = 23$ , which is not divisible by 6 . (iv) Always true

Multiple of 6; 6m Multiple of 9; 9n $6 m + 9 n = 3 (2 m + 3 n)$ Hence multiple of 3 . (v) Sometimes true, the sum of a multiple of 6 and a multiple of 3 is a multiple of 9 .

Multiples of 6 are: $6, 12, 18, 24, 30, \dots$ $6 + 12 = 18$ is a multiple of 9 . $12 + 18 = 30$ is not a multiple of 9 . $18 + 24 = 42$ is not a multiple of 9 . Sometimes true. Multiples of 3 are: 3, 6, 9, 12, 15, 18,... $3 + 6 = 9$ is a multiple of 9 . $6 + 9 = 15$ is not a multiple of 9 . Sometimes true. 4. Find a few numbers that leave a remainder of 2 when divided by 3 and a remainder of 2 when divided by 4 . Write an algebraic expression to describe all such numbers.

Sol. Here, Remainder = 2, Dividend = 3 $∴$ Number $= ($ Quotient × Dividend $) +$ Remainder $= (K \times 3) + 2$ where, $K = 1, 2, 3, \dots$ . . Numbers $= 1 \times 3 + 2 = 3 + 2 = 5$ Numbers $= 2 \times 3 + 2 = 6 + 2 = 8$ Numbers $= 3 \times 3 + 2 = 9 + 2 = 11$ Thus, 5, 8, and 11 are numbers that leave a remainder of 2 when divided by 3 .

Algebraic expression $= 3 K + 2$ Here, Remainder $= 2$ , dividend $= 4$ Number $= 4 K + 2$ , where $K = 1, 2, 3, 4, \dots$ Numbers $= 4 \times 1 + 2 = 4 + 2 = 6$ Numbers $= 4 \times 2 + 2 = 8 + 2 = 10$ Numbers $= 4 \times 3 + 2 = 12 + 2 = 14$ Algebraic expression $= 4 K + 2$ Thus, 6,10 , and 14 are numbers that leave a remainder of 2 when divided by 4 . 5. "I hold some pebbles, not too many, when I group them in 3's, one stays with me. Try pairing them up - it simply won't do. A stubborn odd pebble remains in my view. Group them by 5, yet one's still around, but grouping by seven, perfection is found. More than one hundred would be far too bold. Can you tell me the number of pebbles I hold?"

Sol. The LCM of 3, 5, and 7=3 $\times 5 \times 7 = 105$ [ $∵ 3, 5$ , and 7 are prime numbers]

No. of pebbles = 105 + 1=106,

6. Tathagat has written several numbers that leave a remainder of 2 when divided by 6 . He claims, "If you add any three such numbers, the sum will always be a multiple of 6." Is Tathagat's claim true?

Sol. The expression has been written by Tathagat $= 6 k + 2$ where, $k = 1, 2, 3, 4, 5, 6, \dots$ $6 \times 1 + 2 = 8$ $6 \times 2 + 2 = 14$ $6 \times 3 + 2 = 20$ $6 \times 4 + 2 = 26$ The sum of three numbers $8 + 14 + 20 = 42$ , it is a multiple of 6 . $14 + 20 + 26 = 60$ , it is a multiple of 6 . Yes, Tathagat's claim is true. 7. When divided by 7, the number 661 leaves a remainder of 3, and 4779 leaves a remainder of 5 . Without calculating, can you say what remainders the following expressions will leave when divided by 77? Show the solution both algebraically and visually. (i) $4779 + 661$ (ii) 4779-661

Sol. Given, $661 = K \times 7 + 3$ , where $K = 1, 2, 3, 4, \dots$ and, $4779 = K \times 7 + 5$ Algebraic Method: (i) $4779 + 661$ $4779 = (682 \times 7) + 5$ Remainder $= 5$ $661 = (94 \times 7) + 3$ Remainder $= 3$ $∴ 4779 + 6617 = 5 + 37 = 87$ = 1 = Remainder (ii) 4779-661 $∴ 4779 - 6617 = 5 - 17 = 87$ = 4 = Remainder Visualization Method: (i) $4779 + 661 = (682 \times 7) + 5 + (94 \times 7) + 3$ $= 7 \times (682 + 94) + 5 + 3$ $= 7 \times 776 + 8$ = Divisible by $7 + 87$ $= 1$ , Remainder (ii) $4779 - 661 = (682 \times 7) + 5 - (94 \times 7) - 3$ $= 7 \times (682 - 94) + 5 - 3$ $= 7 \times 588 + 2$ = Divisible by $7 + 2$ $= 2$ , Remainder 8. Find a number that leaves a remainder of 2 when divided by 3 , a remainder of 3 when divided by 4 , and a remainder of 4 when divided by 5 . What is the smallest such number? Can you give a simple explanation of why it is the smallest?

Sol. The expression of a number that leaves a remainder of 2 when divided by 3 .

Number $= 3 K + 2$ $3 \times 1 + 2 = 5$ $3 \times 2 + 2 = 8$ $3 \times 3 + 2 = 11$ $3 \times 4 + 2 = 14$ $3 \times 5 + 2 = 17$ $3 \times 6 + 2 = 20$ The expression of a number that leaves a remainder of 3 when divided by 4 .

Number $= 4 K + 3$ $4 \times 1 + 3 = 7$ $4 \times 2 + 3 = 11$ $4 \times 3 + 3 = 15$ $4 \times 4 + 3 = 19$ The expression of a number that leaves a remainder of 4 when divided by 5 .

Number $= 5 K + 4$ $5 \times 1 + 4 = 9$ $5 \times 2 + 4 = 14$ $5 \times 3 + 4 = 19$ $5 \times 4 + 4 = 24$ Smallest number $= LCM$ of $(3, 4, 5) - 1$ = 60-1 $= 59$ 59 is the smallest number that leaves a remainder of 2 when divided by 3 , a remainder of 3 when divided by 4 , and a remainder of 4 when divided by 5 .

Multiple choice questions

1.Which of the following numbers can be written as the sum of two consecutive numbers? (1) 10 (2) 25 (3) 30 (4) 50

2. The number 36 can be written as the sum of: (1) Two consecutive numbers only (2) Three consecutive numbers only (3) Both two and more than two consecutive numbers (4) Not possible at all

3. If you put plus ( + ) and minus ( - ) signs between $7, 8, 9, 10$ , the result is always: (1) Odd (2) Even (3) Multiple of 5 (4) Prime

4. Which number cannot be expressed as a sum of two or more consecutive numbers? (1) 9 (2) 15 (3) 28 (4) 32

5. Which test checks divisibility by 9 ? (1) Last digit must be 0 or 5 (2) Double the last digit subtract from rest (3) Sum of digits must be multiple of 9 (4) Alternating sum of digits must be multiple of 11

6. The sum of four consecutive integers is always divisible by: (1) 2 only (2) 4 only (3) Both 2 and 4 (4) Neither

7. Which of the following is sometimes true but not always? (1) Adding two odd numbers gives an even (2) Adding two even numbers give an even (3) A number divisible by 6 is also divisible by 12 (4) A multiple of 4 added to another multiple of 4 is a multiple of 4

8. If a number leaves remainder 3 when divided by 4 , then the sum of the next three consecutive numbers will be divisible by: (1) 4 (2) 6 (3) 12 (4) 8

9. Let $a, b, c, d$ be four consecutive integers. Which of the following is always true? (1) $a + b - c - d$ is divisible by 2 (2) $a - b - c - d$ is divisible by 4 (3) $a + b + c + d$ is odd (4) $a \times b \times c \times d$ is odd

10. Which number can be written as the sum of consecutive numbers in exactly 3 different ways? (1) 15 (2) 21 (3) 28 (4) 36

11. The sum of three consecutive odd numbers is divisible by: (1) 2 (2) 3 (3) 6 (4) 9

12. Which of the following expressions is always even for integer values of $a$ and $b$ ? (1) $a + b$ (2) $a \times b$ (3) $(a + b) (a - b)$ (4) $a^{2} + b^{2}$

13. If a number can be expressed as the sum of two consecutive numbers and also as the sum of three consecutive numbers, it must be: (1) A multiple of 2 (2) A multiple of 3 (3) A multiple of 6 (4) An odd prime

14. A number is divisible by both 4 and 6 . Which of the following is it always divisible by? (1) 8 (2) 12 (3) 18 (4) 24

15. Which number when divided by 9 gives remainder 4? (1) 67 (2) 85 (3) 103 (4) 58

16. In the cryptarithm $AB \times C = DDD$ , where $A, B, C$ , and D are distinct digits, what is the value of $A + B + C$ ? (1) 13 (2) 14 (3) 15 (4) 16

17. If 8a75a is a 5-digit number divisible by 11, what is the digit ' $a$ '? (1) 1 (2) 3 (3) 6 (4) 9

18. If the 6-digit number 4x15y2 is divisible by 72 , what is the value of $x + y$ ? (1) 8 (2) 9 (3) 10 (4) 11

19. The product of any three consecutive integers is always divisible by: (1) 4 (2) 5 (3) 6 (4) 9

20. Which of the following statements is always true? (1) The sum of two multiples of 3 is a multiple of 6 . (2) If a number is divisible by 5 , it is also divisible by 10 . (3) The sum of three consecutive integers is a multiple of 3 . (4) The product of two even numbers is a multiple of 8 .

21. The digital root of a number is the same as the remainder obtained when that number is divided by: (1) 2 (2) 3 (3) 5 (4) 9

22. Which of the following numbers is divisible by both 3 and 11 ? (1) 273 (2) 285 (3) 396 (4) 462

A 23. If

$A + A BA$

The possible values of A and $B$ are: (1) $A = 0, B = 2$ (2) $A = 5, B = 1$ (3) $A = 5, B = 0$ (4) $A = 0, B = 5$

24. The number $1 + 6384$ is divisible by (1) 5 (2) 9 (3) 7 (4) 6

25. In the cryptarithm $1 A \times A = 9 A$ , what is the value of A? (1) 5 (2) 6 (3) 7 (4) 8

26. If the number $\overline{517 \times 324}$ is completely divisible by 3 , then the smallest whole number in place of x will be: (1) 0 (2) 1 (3) 2 (4) None of these

27. If the number $\overline{481 \times 673}$ is completely divisible by 9 , then the smallest whole number in place of x will be: (1) 2 (2) 5 (3) 6 (4) 7

28. If the number $\overline{27215 \times 2}$ is completely divisible by 11 , then the smallest whole number in place of x will be: (1) 3 (2) 2 (3) 1 (4) 5

29. If the number $\overline{91876 \times 2}$ is completely divisible by 8 , then the smallest whole number in place of $x$ will be: (1) 1 (2) 2 (3) 3 (4) 4

30. Which one of the following numbers is completely divisible by 99 ? (1) 3572404 (2) 135792 (3) 913464 (4) 114345

31. Statement-1: The sum of four consecutive integers is always even.

Statement-2: The sum of five consecutive integers is always a multiple of 5 . (1) Statement-1 is true and statement-2 is false. (2) Statement-1 is false and statement-2 is true. (3) Both statements are true. (4) Both statements are false.

32. Statement-1: Any number that is divisible by 9 will also be divisible by 3 . Statement-2: Any number that is divisible by 3 will also be divisible by 9 . (1) Statement-1 is true and statement-2 is false. (2) Statement-1 is false and statement-2 is true. (3) Both statements are true. (4) Both statements are false.

33. Statement-1: The sum of two even numbers is always a multiple of 4 . Statement-2: The sum of two odd numbers is always even. (1) Statement-1 is true and statement-2 is false. (2) Statement-1 is false and statement-2 is true. (3) Both statements are true. (4) Both statements are false.

34. Statement-1: If a number leaves a remainder of 2 when divided by 3 , it will also leave a remainder of 2 when divided by 6 . Statement-2: If a number is divisible by 6 , it will also be divisible by both 2 and 3 . (1) Statement-1 is true and statement-2 is false. (2) Statement-1 is false and statement-2 is true. (3) Both statements are true. (4) Both statements are false.

35. Statement-1: The product of three consecutive integers is always divisible by 6 . Statement-2: The product of four consecutive integers is always divisible by 12 . (1) Statement-1 is true and statement-2 is false. (2) Statement-1 is false and statement-2 is true. (3) Both statements are true. (4) Both statements are false.

Fill in the Blanks 36. The sum of three consecutive odd numbers, such as 11,13 , and 15 , is always divisible by $____$ .

37. Placing plus and minus signs in between any four consecutive numbers always gives results with the same $____$ .

38. The difference between the squares of 51 and 50 is always an $____$ number.

39. The sum of six consecutive numbers, for example $8 + 9 + 10 + 11 + 12 + 13$ , is always divisible by $____$ .

True or False

40. The sum of four consecutive integers is always even.

41. Every odd number can be expressed as the sum of exactly two consecutive natural numbers.

42. The sum of two even numbers is sometimes odd.

43. A number divisible by 12 is also divisible by 3 and 4 .

44. The product of three consecutive numbers is always divisible by 9 .

The Pebble Riddle

1.Solve this riddle to find the secret number of pebbles! I'm made of digits, each tiniest and odd, No shared ground with root #1-how odd! My digits count, their sum, my root- All point to one bold number's pursuit- The largest odd single-digit I proudly claim. What's my number? What's my name?

ANSWER KEY

Multiple choice questions

Question	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Answer	2	3	2	4	3	3	3	1	1	1	2	3	3	2	2
Question	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Answer	4	3	4	3	3	4	4	2	1	2	3	4	1	3	4
Question	31	32	33	34	35
Answer	1	1	2	2	3

Fill in the Blanks 36.3 37. Parity 38. Odd 39.6

True of False 40. True 41. True 42. False 43. True 44. False

The Pebble Riddle 45.135

3.0Key Concepts of Chapter 5 Exercise 5.1

Numbers in General Form: Representing a two-digit number ab as 10a + b and a three-digit number abc as 100a + 10b + c.
Letters for Digits: Each letter in a puzzle stands for a single digit (0–9). The first digit of a number cannot be zero.
Cryptarithms: Mathematical puzzles where digits are replaced by letters. The goal is to find which letter represents which digit based on standard arithmetic rules.
Logic of Addition and Multiplication: Using carry-over and units-place rules to narrow down the possible values for a letter.

4.0NCERT Solutions for Class 8 Maths Chapter 5 Number Play : All Exercises

NCERT Solutions for Class 8 Maths Chapter 5 Number Play - Exercise 5.1

NCERT Solutions for Class 8 Maths Chapter 5 Number Play - Exercise 5.2

NCERT Solutions for Class 8 Maths Chapter 5 Number Play - Exercise 5.3

NCERT Solutions for Class 8 Maths Chapter 5 Number Play - Exercise 5.4

5.0Benefits of NCERT Solutions for Class 8 Maths Chapter 5 Exercise 5.1

Conceptual Clarity: Helps students understand that numbers are more than just symbols; they follow strict place-value patterns.
Logical Reasoning: Encourages "trial and error" within a logical framework to solve puzzles.
Exam-Oriented: Focuses on the single-digit constraints that are key to scoring in this chapter.
Error Prevention: Teaches students to always check their final letter values back in the original sum.