NCERT Solutions for Class 8 Maths Chapter 5 Number Play - Exercise 5.3
Mastering complex number patterns is simple with our NCERT Solutions for Class 8 Maths Ch 5: Playing with Numbers – Exercise 5.3. This exercise expands your toolkit to include the divisibility rules for 2, 5, 10, and the "alternating sum" rule for 11. You will also learn the logic behind constructing 3x3 Magic Squares where every row, column, and diagonal adds up to the same constant.
Class 8 Math Lessons (Ch 5): Learn to Solve Advanced Number Puzzles with Ease! The solutions to Exercise 5.3 will be presented in a step-by-step approach to help students identify multiples of 11 and complete magic grids. These solutions are in accordance with the CBSE Guidelines to improve logical reasoning and pattern recognition. Additionally, these solutions build the foundation for number theory and computer science algorithms.
1.0Download Class 8 Maths Chapter 5 Number Play Ex 5.3 NCERT Solutions PDF
Chapter 5 of our simple NCERT Solutions for Class 8 Maths You may download the PDF for Exercise 5.3. Ideal for fast reference and offline study.
2.0 Detailed NCERT Class 8 Maths Chapter 5 Solutions of Exercise 5.3
1.The digital root of an 8-digit number is 5 . What will be the digital root of 10 more than that number?
Sol. Consider the 8-digit number 80000006.
The digital root of 80000006=8+0+0+
0+0+0+0+6
= 14
=1+4
=5
10 more than 80000006=80000006+10
=80000016
The digital root of 80000016=8+0+0+
0+0+0+1+6
= 15
=1+5
=6
Thus, the digital root of 10 more than 80000006 is 6 .
2. Write any number. Generate a sequence of numbers by repeatedly adding 11 . What would be the digital roots of this sequence of numbers? Share your observations.
Sol. Consider the number =40
The sequence of numbers by repeatedly adding 11 are 40,51(40+11),62(51+ 11), 73(62+11),84(73+11),95(84+ 11), 106(95+11),117(106+11), 128(117+11),139(128+11), etc.
The digital roots of this sequence of numbers are:
40=4+0=4;
51=5+1=6;
62=6+2=8;
73=7+3=10=1+0=1;
84=8+4=12=1+2=3;
95=9+5=14=1+4=5;
106=1+0+6=7;
117=1+1+7=9;
128=1+2+8=11=1+1=2;
139=1+3+9=13=1+3=4,... etc.
Thus, the digital roots of this sequence of numbers are 4,6,8,1,3,5,7,9,2,4,…….
Observations:
The digital roots are 4,6,8,1,3,5,7,9,2, 4,......
This sequence starts repeating after 9 steps.
So the digital roots form a cycle: 4,6,8,1, 3,5,7,9,2,4,…….
3. What will be the digital root of the number 9a+36b+13 ?
Sol. First Method:
The digital root of the number 9a+36 b+13=9a+36 b+9+4
=9(a+4 b+1)+4
=9+4
=13[∵ The digital root of multiples of 9 is always 9.]
=1+3=4
Thus, the digital root of the number 9a + 36 b+13 will be 4 .
Second Method:
We have 9a+36b+13
Here, a and b are integers
Put a=1, b=1,
9a+36b+13=9×1+36×1+13
=9+36+13
=58
The digital root of 58=5+8=13=1+3 = 4
Put a=2,6=3,
9a+36b+13=9×2+36×3+13
=18+108+13
= 139
The digital root of 139=1+3+9=13=1
+3=4
Thus, the expression 9a + 36b + 13 always has a digital root of 4 .
4. Make conjectures by examining if there are any patterns or relations between
(i) the parity of a number and its digital root.
(ii) the digital root of a number and the remainder obtained when the number is divided by 3 or 9 .
Sol. Consider the pattern: 8,16,24,32,40,…….
(i) 8=8= digital root, parity → even
16=1+6=7= digital root, parity → odd
24=2+4=6= digital root, parity → even
32=3+2=5= digital root, parity → odd
40=4+0=4= digital root, parity → even
(ii) Divided by 3
8÷3⇒2, Remainder
24÷3⇒0, Remainder
32÷3⇒2, Remainder
40÷3⇒1, Remainder
Divided by 9
8÷9⇒8, Remainder
24÷9⇒6, Remainder
32÷9⇒5, Remainder
40÷9⇒4, Remainder
3.0Key Concepts of Chapter 5 Number Play Exercise 5.3
- Divisibility by 10 and 5: Focuses on the unit's digit (0 for 10; 0 or 5 for 5).
- Divisibility by 2: Focuses on the unit's digit being even (0, 2, 4, 6, 8).
- Divisibility by 11: A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.
- Magic Squares: A square grid where the sum of numbers in each row, each column, and both main diagonals is the same (the Magic Constant).
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 5.3
- Conceptual Clarity: Provides a logical explanation for the "jump-step" method used for 11.
- Brain Training: Magic squares improve mental addition and spatial reasoning.
- Exam-Oriented: Covers the "find the missing digit for divisibility by 11" which is a common competitive exam question.