NCERT Solutions for Class 8 Maths Chapter 5 Number Play - Exercise 5.2
Mastering the shortcuts to identify factors is simple with our NCERT Solutions for Class 8 Maths Ch 5: Playing with Numbers – Exercise 5.2. This exercise focuses on the "Divisibility Rules" for 3 and 9. By using the generalized form of numbers, you will learn why adding digits tells us if a number is divisible by these specific values.
Class 8 Math Lessons (Ch 5): Learn to Solve Divisibility Puzzles with Ease! The solutions to Exercise 5.2 will be presented in a step-by-step approach to help students find missing digits in large numbers. These solutions are in accordance with the CBSE Guidelines to provide the best path for improving speed in mental calculations. Additionally, these solutions build the foundation for simplifying fractions and finding prime factors efficiently.
1.0Download Class 8 Maths Chapter 5 Number Play Ex 5.2 NCERT Solutions PDF
Our easy-to-follow NCERT Solutions for Class 8 Maths Chapter 5 Exercise 5.2 are available for download as a PDF. Perfect for offline study and quick reference.
2.0 Detailed NCERT Class 8 Maths Chapter 5 Solutions of Exercise 5.2
1.Find, without dividing, whether the following numbers are divisible by 9 .
(i) 123
(ii) 405
(iii) 8888
(iv) 93547
(v) 358095
Sol. If the sum of the digits of a number is divisible by 9, then the number is divisible by 9 .
(i) Sum of the digits = 1+2+3=6, is not divisible by 9 .
Thus, 123 is not divisible by 9 .
(ii) Sum of the digits =4+0+5=9, is divisible by 9 .
Thus, 405 is divisible by 9 .
(iii) Sum of the digits =8+8+8+8=32, is not divisible by 9 .
Thus, 8888 is not divisible by 9 .
(iv) Sum of the digits =9+3+5+4+7= 28, is not divisible by 9 . Thus 93547 is not divisible by 9 .
(v) Sum of the digits =3+5+8+0+9+5=30, is not divisible by 9 .
2. Find the smallest multiple of 9 with no odd digits.
Sol. Multiples of 9=9,18,27,36,…,288, ____ The smallest multiple of 9 with an odd digit is 9.
The smallest multiple of 9 that can be formed by summing even digits is 18 (since 9 is odd).
Thus, the smallest multiple of 9 with no odd digits is 288.
3. Find the multiple of 9 that is closest to the number 6000.
Sol. Given, 6000
Sum of the digits =6+0+0+0=6
We know that, if the number is divisible by 9 , then the sum of the digits is divisible by 9 .
If we add 3 to the number 6000.
6000+3=6003, it is divisible by 3 .
Thus, the multiple of 9 that is closest to the number is 6003.
4. How many multiples of 9 are there between the numbers 4300 and 4400 ?
Sol. The multiples of 9 are there between the numbers 4300 and 4400 are 4302, 4311, 4320, ____ 4392
The number of multiples of 9=
Last term - First term Difference
=94392−4302+1
=990+1
=10+1
=11
Thus, the multiples of 9 are 11.
Very short answer type questions
1.The sum of three consecutive integers is 48. Find the numbers.
2. Find the digital root of the following numbers:
(a) 248
(b) 7365
(c) 90708
Then verify if the digital root of each number matches its remainder when divided by 9 .
3. If a number n=5p+2, where p is any whole number, then what will be the remainder when n is divided by 5 ? Explain your answer.
4. What is the digital root of the algebraic expression 27p+18q+33, where p and q are any positive integers.
Short answer type questions
5.Let the greatest of five consecutive even numbers be p.
Express all five numbers in terms of p and find their average.
6. A number when divided by 7 leaves remainder 3, and when divided by 5 leaves remainder 2 .
Find the smallest such number greater than 50 .
7. Show algebraically that the sum of two multiples of 6 is always a multiple of 6 . Can you find two multiples of 6 whose sum is also a multiple of 12 ? Give examples.
8. A student claims that if a number is divisible by both 3 and 4 , it must also be divisible by 12 .
Is the claim always true? Prove or give a counterexample.
9. Write any three numbers that leave remainder 1 when divided by 4 and remainder 2 when divided by 5 .
10. Find all three-digit numbers that are divisible by 9 and also by 11 .
Explain your reasoning.
11. A student observes that when a number is increased by 27, its digital root does not change.
Can this happen?
Explain algebraically why or why not.
12. Write five multiples of 48 between 30,000 and 30,300. Explain your approach in detail.
13. Find all two-digit numbers that are multiples of 4 but not of 8 . Explain why the pattern repeats every 8 numbers.
14. Express four consecutive odd numbers in terms of n and find their sum and product. State whether the product will always be divisible by 8 .
15. Consider the pattern:
10, 20, 30, 40, 50 ____
(i) Find the digital root and parity (even/odd) of each number.
(ii) Find the remainders when these numbers are divided by 4 and by 7 . Then write any patterns or relations you observe.
16. Choose any three consecutive multiples of 7. Prove that their sum is always divisible by 21 and discuss whether this pattern will hold for multiples of 5 or 9.
17. A certain integer has a digital root of 7. If this integer is tripled, and then increased by 8 , determine the resulting digital root of the new number.
Long answer type questions
18. Find all two-digit numbers that are divisible by 9 and have the sum of digits equal to 9.
How many such numbers exist? Show your reasoning clearly.
19. A number leaves a remainder of 1 when divided by 4 , a remainder of 2 when divided by 5, and a remainder of 3 when divided by 6 .
Find the smallest such number and explain your reasoning using a step-bystep method.
20. Check whether the following statements are Always True, Sometimes True, or Never True. Justify your answers with examples or algebraic reasoning.
(i) The product of two consecutive integers is even.
(ii) The sum of two multiples of 9 is a multiple of 18.
(iii) The sum of two even numbers is odd.
21. The sum of three numbers is 72. The first is twice the second, and the third is four less than the second.
Find all three numbers.
Then, check whether each number is divisible by 3 .
22. A four digit number is written in the form 4ab6, where a and b are digits (0-9). Find all possible values of a and b such that the number is divisible by both 3 and 4.
23. If the number 37a45b (a and b are single digits) is a multiple of 18, list all possible pairs ( a,b ). Explain your reasoning.
24. Prove algebraically that the difference between the squares of any two consecutive numbers is always odd. Illustrate your proof with two examples.
Direction (Q. 25 to Q.30):
Solve these cryptarithms:
Each letter represents one digit.
Find A,B,C and verify your answer.
25. AB×3=CAA
26. A B
27. A B
28.
A1+1 B B0
29.
×CA3 A B B
30.
12 A+6 A B A09
31. Given that the number 35a64 is divisible by 3 , where a is a digit, what are the possible values of a ?
32. If x is a digit such that the number 18×71 is divisible by 3 , find possible values of x .
33. If x is a digit of the number 66784x such that it is divisible by 9 , find possible values of x .
34. If 3×2 is a multiple of 11 , where x is a digit, what is the value of x ?
35. If x denotes the digit at hundreds place of the number 67×19 such that the number is divisible by 11 . Find all possible values of x .
ANSWER KEY
Very short answer type questions
- 15, 16, 17
- (a) 5
(b) 3
(c) 6
3.2
- 6
Short answer type questions
5. p−8,p−6,p−4,p−2,p; Average =p−4
6. 52
7. 12, 36
8. Yes, Always true
9. 17, 37, 57
10. 198, 297, 396, 495, 594, 693, 792, 891, 990
11. Yes
12. 30000, 30048, 30096, 30144, 30192
13. 12,20,28,36,44,52,60,68,76,84,92
14. Sum =4n+12; Product =n(n+2)(n+4)(n+6)
15. (i) 1,2,3,4,5........ Parity - even
(ii) When divided by 4 , the remainder are 2,0,2,0,2,0…….
When divided by 7 , the remainder are 3,6,52,1 pattern repeats after 7 steps.
16. Yes, the pattern holds for multiples of 5 and multiples of 9 .
17. 2
Long answer type questions
18. 18,27,36,45,54,63,72,81,90; Count - 9
19. 57
20.
(i) Always true
(ii) Sometimes true
(iii) Never true
21. 38, 19 and 15; No, No, Yes
22. For b=1,3,7,9a∈{1,4,7}; For b=5a∈{0,3,6,9}
23. (8,0),(6,2),(4,4),(2,6),(0,8),(9,8)
25. A=4, B=8,C=1
26. C=2, B=0, A=5
27. A=7, B=4
28. B=9,A=7
29. A=5, B=0,C=1
30. A=8 and B=1
31. 0,3,6,9
32. 1, 4, 7
33.5
34.5
35.4
e
3.0Key Concepts of Chapter 5 Number Play Exercise 5.2
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Generalized Form Proof: Understanding that abc = 100a + 10b + c = 99a + 9b + (a + b + c), which proves why the sum of digits matters.
- Finding Missing Digits: Solving for an unknown digit x or y when the number is already known to be a multiple of 3 or 9.
4.0NCERT Solutions for Class 8 Maths Chapter 5 Number Play : All Exercises
5.0Benefits of NCERT Solutions for Class 8 Maths Chapter 5 Exercise 5.2
- Conceptual Clarity: Explains the logic behind divisibility rules rather than just memorizing them.
- Problem-Solving Skills: Teaches how to set up simple algebraic equations to find unknown digits.
- Accuracy: Helps students identify all possible values for a variable, ensuring they don't lose marks for incomplete answers.