Experts have designed these Class 8 Maths Notes and Chapter 5 Number Play Class 8 Notes for effective learning.
Class 8 Maths Chapter 5 Notes Number Play
Class 8 Maths Notes Chapter 5 – Class 8 Number Play Notes
In our previous classes, we have explored the fascinating world of numbers – how they behave, how they are built, and how they interact. We have learned about even and odd numbers, discovered what makes a number a multiple or a factor, and learned simple ways to check if a number can be divided by another without leaving a remainder. We have also played with patterns and puzzles to sharpen our number sense.
In this chapter, we will take those skills even further. We will revisit divisibility rules, explore parity in a new light, and challenge our minds with cryptarithms – clever puzzles where letters stand in for digits.
→ Numbers in General Form:
- Generally, a 2-digit number “ab” can be written as 10 a+b, where a and b are whole numbers and a ≠ 0.
- Generally, a 3-digit number “abc” can be written as 100 a+10 b+c, where a, b, and c are whole numbers and a ≠ 0.
→ Properties of Divisibility:
- If a is divisible by b, then all the multiples of a are divisible by b.
- If a is divisible by b, then a is divisible by all the factors of b.
- If a divides m and a divides n, then a divides m+n and m-n.
- If a is divisible by b and is also divisible by c, then a is divisible by the LCM of b and c.
→ Divisibility Rules: A number is divisible by:
- 2, if its ones digit is an even number.
- 10, if its ones digit is 0.
- 5, if the ones digit of a number is either 5 or 0.
- 3 and 9 if the sum of digits is divisible by 3 and 9 respectively.
- 11, if the difference between the sum of its digits in odd places and the sum of its digits in even places is either 0 or a multiple of 11.
→ Digital Roots: The single-digit value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute the digit sum. The process continues until a single-digit number is reached. The digital root of a number is the remainder obtained when a number is divided by 9.
→ Cryptarithms is a type of mathematical puzzle in which the digits are replaced by letters of the alphabet or other symbols. The solution involves finding the original digits. Making and solving such puzzles are known as Cryptarithmetic or Cryptarithms. While solving Cryptarithms, some conventions made are as follows:
- (a) Each letter or symbol in a puzzle stands for just one digit and is represented by just one letter.
- (b) The first digit of a number cannot be zero.
- (c) When letters are replaced by their digits, the resultant arithmetical operation must be correct.
This chapter introduces the patterns in numbers through the study of consecutive sums, parity and multiples of 4. It also covers the rules of divisibility by 2, 3,4, 5, 6, 7, 8, 9, 10, 11 for faster and accurate calculations.
Sum of Consecutive Numbers
A number can be expressed as the sum of two or more consecutive numbers.
e.g. 7 can be written as 3 + 4 and 12 can be written as 3 + 4 + 5.
Parity in Expressions with Consecutive Numbers
Parity The concept of parity refers to whether a number is even or odd.
Four consecutive numbers When placing ‘+’ and ’ signs between four consecutive numbers a, b, c and d. All possible expressions yield results with the even parity, e.g. Let the four numbers are 3, 4, 5, 6.
Now, 3 + 4 + 5+ 6 = 18 (even)
3 + 4 + 5 – 6 = 6 (even)
3 + 4 – 5 + 6 = 8 (even) and so on.
All results are even.
Properties of Even and Odd Numbers
Addition and Subtraction Rules
- Even ± Even = Even
- Odd ± Odd = Even
- Even ± Odd = Odd
Multiplication Rules
- Even × Any number = Even
- Odd × Odd = Odd
Properties of Sum of Even Numbers and Multiples of 4
The following two cases arise :
1. Even numbers which are multiples of 4
e.g. 4, 8, 12, ………..
2. Even numbers which are not multiples of 4
e.g. 2, 6, 10, ………….
Properties
(i) The sum of two multiples of 4 is always a multiple of 4.
(ii) The sum of two even numbers that are not multiples of 4 is a multiple of 4.
(iii) The sum of a multiple of 4 and an even number that are not divisible by 4, is even but not a multiple of 4.
Test for Divisibility
To check the divisibility of one number by other, we normally perform actual divison and see whether remainder is zero or not.
Test of Divisibility by 2
A number is divisible by 2, if it has any of the digits 0, 2, 4, 6 or 8 in its ones place.
e.g. The numbers 30, 52, 144, 246,448 are divisible by 2.
Test of Divisibility by 3
A number is divisible by 3, if the sum of its digits is divisible by 3.
Test of Divisibility by 4
A number is divisible by 4, if the number formed by its last two digits (i.e. ones and tens) is divisible by 4.
Test of Divisibility by 5
A number is divisible by 5, if it has either 0 or 5 in its ones place.
e.g. Each of the number 60, 225, 625 is divisible by 5.
Test of Divisibility by 6
A number is divisible by 6, if the number is divisible by 2 and 3 both.
e.g. Each of the number 18,24, 36 is divisible by 6.
Test of Divisibility by 7
A number is divisible by 7, if the difference between twice the ones digit and the number formed by the other digits is either 0 or a multiple of 7.
Test of Divisibility by 8
A number is divisible by 8, if the number formed by the last three digits is divisibile by 8.
Test of Divisibility by 9
A number is divisible by 9, if the sum of its digits is divisible by 9.
Test of Divisibility by 10
A number is divisible by 10, if it has 0 in its ones place.
e.g. Each of the number 100, 200, 300, 450, is divisible by 10.
Test of Divisibility by 11
A number is divisible by 11, if the difference between the sum of the digits at odd places (from the right to left) and the sum of the digits at even places (from the right to left) of the number is either 0 or divisible by 11.
General Divisibility Rules
1. If two given numbers are divisible by a number, then their sum is also divisible by that number, e.g. 24 and 30 are divisible 6 and the sum of these numbers
i.e. 24 + 30 = 54 is also divisible by 6.
2. If two given numbers are divisible by a number, then their difference is also divisible by that number, e.g. 16 and 36 are divisible by 4 and their difference
i. e. 36 -16 = 20 is also divisible by 4.
3. If a number N is divisible by another number, then all multiples of N is divisible by that number.
e.g. If 4 is divisible by 2, then 8, 12, 16, ……….are also divisible by 2.
4. If a number is divisible by another number, then it is divisible by each of the factors of that number, e.g. 225 is divisible by 15.
Factors of 15 are 1, 3 and 5.
∴ 225 is also divisible by 3 and 5.
5. If a number is divisible by two numbers, then that number is also divisible by their LCM.
e.g. 12 is divisible by both 2 and 3.
LCM of 2 and 3 = 6
∴ 12 is divisible by 6.
Remainder
The remainder is the value left after the division. If a number (divided) is not completely divisible by another number (divisor), then we are left with a number once the division is done. This value is called the remainder, e.g. 11 is not exactly divisible by 3.
Since, the closest value, we can get 3 × 3 = 9
∴Remainder = 11 – 9 = 2
Digital Root
The digital root of a number is the single digit obtained by adding the number successively,
e.g. Find the digital root of 539.
Here, sum of digits of 539 = 5 + 3+ 9 = 17
and sum of digits of 17 = 1 + 7 = 8
So, digital root of 539 = 8.
Cryptarithms
A cryptarithm is a mathematical puzzle where letters or symbols represent digits in an arithmetic equation. The goal is to find the unique digit (0-9) that each letter represents to make the equation true.
e.g. Find the value of Q.
In first column, Q + 3 = 1, means addition of these numbers equals a number whose units digit is 1.
So, Q should be 8.
∴ 8 + 3 = 11
On putting this value, we get
So, Q = 8
Note Two rules are followscLaibfte doing such puzzles.
(i) Each letter in the puzzle must stand for just one digit. Each digit must be represented by just one letter.
(ii) The first digit of a number cannot be zero. Thus, we write . the dumber 63 as 63, not as 063 or 0063.