Class 8 NCERT Solutions - Chapter 16 Playing with Numbers - Exercise 16.2

Exercise 16.2 of Chapter 16 (Playing with Numbers) in the Class 8 NCERT Mathematics textbook focuses on number puzzles and games. This exercise aims to develop students' logical thinking, problem-solving skills, and numerical reasoning through engaging mathematical puzzles.

Question 1.If 21y5 is a multiple of 9, where y is a digit, what is the value of y?

Solution: 

According to the divisibility rule of 9,the sum of all digits should be a multiple of 9

Sum of the digits of 21y5 = 2 + 1 + y + 5 = 8 + y

(8 + y) ÷ 9 = 1

8 + y = 9

y = 9 - 8 = 1

Hence, the required value of y = 1.

Question 2. If 31z5 is a multiple of 9, where z is a digit, what is the value of z? You will find that there are two answers for the last problem. Why is this so?

Solution:

According to the divisibility rule of 9, the sum of all digits should be a multiple of 9

Sum of the digits of 31z5 = 3 +1 + z + 5 = 9 + z  

9 + z = 9

z = 0

9 + z = 18

z = 9

Hence, 0 and 9 are two possible answers.

Question 3. If 24x is a multiple of 3, where x is a digit, what is the value of x?

(Since 24x is a multiple of 3, its sum of digits 6 + x is a multiple of 3; so 6 + x is one of these numbers: 0, 3, 6, 9, 12, 15, 18, ... . But since x is a digit, it can only be that 6 + x = 6 or 9 or 12 or 15. Therefore, x = 0 or 3 or 6 or 9. Thus, x can have any of four different values).

Solution:

Let us assume that 24x is a multiple of 3

According to the divisibility rule of 3,the sum of all digits should be a multiple of 3

Sum of the digits of 24x = 2 + 4 + x =  6 + x

6 + x = 6 if x = 0

6 + x = 9 if x = 3

6 + x = 12 if x = 6

6 + x = 15 if x = 9

Hence, x can have any of the four values

Question 4. If 31z5 is a multiple of 3, where z is a digit, what might be the values of z?

Solution:

According to the divisibility rule of 3,the sum of all digits should be a multiple of 3

Sum of the digits of 31z5 = 3 + 1 + z + 5 = 9 + z

9 + z = 9 if z = 0

9 + z = 12 if z =3

9 + z = 15 if z = 6

9 + z = 18 if z = 9

Hence, 0, 3, 6 and 9 are four possible values

Summary

Exercise 16.2 of Chapter 16 on Playing with Numbers provides students with an engaging and challenging set of number puzzles and games designed to enhance their mathematical thinking. This exercise goes beyond routine calculations, encouraging students to apply their knowledge of arithmetic, algebra, and number properties in creative and logical ways. The problems range from identifying patterns in sequences to solving cryptarithmetic puzzles and completing magic squares. By working through these diverse challenges, students develop crucial skills such as pattern recognition, logical reasoning, and systematic problem-solving. The exercise also promotes mental flexibility, as students must adapt their thinking to different types of puzzles. This approach not only reinforces mathematical concepts learned in earlier chapters but also prepares students for more advanced problem-solving in higher mathematics. Additionally, the gamified nature of these problems helps to build enthusiasm for mathematics, showing students that numbers can be fun and intriguing beyond just routine calculations.