Exercise 16.1 in Chapter 16 of the Class 8 NCERT Mathematics textbook focuses on number patterns, divisibility rules, and general properties of numbers. This exercise builds upon basic arithmetic concepts and introduces students to more advanced number theory ideas. Students will learn to recognize patterns, apply divisibility rules, and explore various properties of integers.
Question 1.
3 A
+ 2 5
--------
B 2
--------
Solution:
A + 5 we get the unit digit as 2
if A = 7 , we get 7 + 5 = 12
The value of B = 1(carry) + 3 + 2 = 6
Hence the addition is
3 7
+ 2 5
--------
6 2
--------
Hence A = 7 and B = 6
Question 2.
4 A
+ 9 8
------------
C B 3
------------
Solution:
A + 8 we get the unit digit as 3
if A = 5, we get A + 8 = 13
The values of B and C = 1(carry) + 4 + 9 = 14
1 is taken as carry
Hence the addition is
4 5
+ 9 8
------------
1 4 3
------------
Hence A = 5, B = 4 and C = 1
Question 3.
1 A
X A
-------------
9 A
--------------
Solution:
A x A = A if A = 1 or 6
For A = 1
1 x 1 = 1 which is not equal to 9
For A = 6
6 x 6 = 36
1 x 6 = 6
Hence the multiplication is
1 6
X 6
-------------
9 6
--------------
Hence A = 6
Question 4.
A B
+ 3 7
--------
6 A
--------
Solution:
A + 3 = 6
A = 6 - 3 = 3 = 2 + 1(carry)
A = 2
B + 7 we get the unit digit as 2
if B = 5 we get 5 + 7 = 12
Hence the addition is
2 5
+ 3 7
--------
6 2
---------
Hence A = 2 and B = 5
Question 5.
A B
X 3
-------------
C A B
-------------
Solution:
B x 3 = B
B = 0
A x 3 = CA
if A = 5 then 5 x 3 = 15
and C = 1
Hence the multiplication is
5 0
X 3
-------------
1 5 0
-------------
Hence A = 5 , B = 0 and C = 1
Question 6.
A B
x 5
-------------
C A B
-------------
Solution:
B x 5 = B
B = 0 or 5
A x 5 = CA
if A = 5 then 5 x 5 = 25
and C = 2
Only possible when B = 0
Hence the multiplication is
5 0
X 5
-------------
2 5 0
-------------
Hence A = 5 , B = 0 and C = 2
Question 7.
A B
× 6
------------
B B B
------------
Solution:
B x 6 = B
if B = 4
4 x 6 = 24
where 2 is carry
A x 6 = BB
if A = 7
7 x 6 = 42
Hence multiplication is
7 4
× 6
------------
4 4 4
-------------
Hence A = 7 and B = 4
Question 8.
A 1
+ 1 B
-----------
B 0
-----------
Solution:
1 + B we get unit digit as 0
if B = 9
1 + 9 = 10
where 1 is the carry
1(carry) + A + 1 = B = 9
A + 2 = 9
A = 9 - 2 = 7
Hence addition is
7 1
+ 1 9
-----------
9 0
-----------
Hence A = 7 and B = 9
Question 9.
2 A B
+ A B 1
---------------
B 1 8
---------------
Solution:
B + 1 we get the unit digit as 8
if B = 7
7 + 1 = 8
A + 7 we get the tens digit as 1
if A = 4
4 + 7 = 11
where 1 is the carry
1(carry) + 2 + 4 = 7
Hence addition is
2 4 7
+ 4 7 1
---------------
7 1 8
----------------
Hence A = 4 and B = 7
Question 10.
1 2 A
+ 6 A B
----------------
A 0 9
----------------
Solution:
A + B = 9
if A = 8 and B =1
8 + 1 = 9
2 + A = 0
if A = 8
2 + 8 = 10
where 1 is the carry
1(carry) + 1 + 6 = A = 8
Hence the addition is
1 2 8
+ 6 8 1
----------------
8 0 9
----------------
Hence A = 8 and B = 1
Summary
Chapter 16 of the Class 8 NCERT Mathematics textbook, titled "Playing with Numbers," delves into various aspects of number theory and pattern recognition. The chapter begins by reinforcing students' understanding of basic arithmetic operations and then introduces more complex concepts related to number properties. Students explore arithmetic sequences, geometric progressions, and other number patterns, learning to identify rules governing these sequences. The chapter also covers divisibility rules for various numbers, helping students quickly determine whether a number is divisible by another without performing the division. Concepts of factors, multiples, and prime factorization are revisited and extended. The chapter introduces the ideas of HCF (Highest Common Factor) and LCM (Least Common Multiple) and their applications in solving real-world problems. Through various examples and exercises, students develop skills in logical thinking, pattern recognition, and problem-solving. The chapter emphasizes the importance of understanding number properties in fields such as cryptography, computer science, and data analysis. By the end of the chapter, students should be able to confidently work with number patterns, apply divisibility rules, and solve problems involving factors and multiples. This knowledge forms a crucial foundation for more advanced mathematical concepts in algebra, number theory, and discrete mathematics.