Game of Numbers – Playing with Numbers | Class 8 Maths
Last Updated : 13 Feb, 2023
We use numbers in our day-to-day life. We buy everything with money and measure its quantity with the help of Numbers only. Therefore Numbers play a very significant role in our life.
The common Representation of a Number is as follows:
The general form of a two-digit number is ab=(10 × a)+b
Here ab is the usual form and (10 × a)+b is the generalized form of the two-digit number.
- 48 = 10 × 4 + 8
- 67 = 10 × 6 + 7
The general form of a three-digit number is abc =(a × 100) + (b × 10) + (c × 1)
Here abc is the usual form of the number and (a × 100) + (b × 10) + (c × 1) is the generalized form of the 3-digit number.
- 237 = 2 × 100 + 3 × 10 + 7 × 1
- 437= 4 × 100 + 3 × 10 + 7 × 1
Game With Numbers
Games with numbers are the tricks that are applicable to all the two digits and three-digit numbers.
1. Reversing Two – Digit Number
To Reverse a two-digit number we have to interchange the digits ie 35 when Reversed 53, ones digit place to tens digit place and tens digit place to ones digit place. Now add them up and divide the result obtained by 11 we observe that there is no remainder left.
By adding we get 88 by dividing by 11 the remainder is 0.
2. Reversing Three – Digit Number
Let us consider a three-digit number 345 on reversing it we get 543. Subtract the larger number from the smaller number and divide the result obtained by 99. We get the remainder as 0.
543 – 345 = 198, we get quotient as 2 when divided by 99.
3.Forming a Three Digit Number From The Given Three Digits
If the three digits are given we can generate 6 combinations of numbers
Eg: 3,7,8 the numbers can be generated using three digits as follows 378, 387, 873, 837, 783, 738.
Let us think of a number ie 754
Form a number such that the last digit of a number should become first digit. Not changing the remaining digits. Ie 7 and 5.
We get 475 and form a number from the number we get, shift the last digit to the place of the first digit and the number we get is 547.now sum up the original numbers along with newly formed numbers and divide the result by 37. We get the remainder as 0.
Letter For Digits
2 is multiplied by a number = 60. What is that number ? The number is 30. We don’t know that number; we assume that number as a variable and what value to be assigned to that variable in order to get the result as 60. Variable is defined as an unknown quantity. No definite value and it keeps changing. It is a puzzle solving type in which we have to find a value for a specified letter. The value to a letter has to have only one digit. If there are multiple letters in a puzzle same values cannot be assigned to multiple letters. The value to a letter cannot be zero if the letter is in the starting position. Eg: 3c + 45 = k4 here what are the values of c and k?. c =9 and k = 8
Putting values of c and k we get, 39 + 45 = 84
9+5=14,1 carry last digit is 4 by substituting 9 as a value to the letter or a variable c we get the last digit as 4.
Let us consider an example A0 x B0=1500 what are the values of A, B? A =3 OR A = 5 B = 5 OR B = 3
Its very simple by looking at this we can determine the values of A and B
i.e 50 × 30 or 30 × 50.
You can try more such interesting examples and try it. It’s not only fun and enjoyable but also its sharpness our brain by solving such problems.
Introduction To Divisibility Rule
Suppose I have 537 chocolates and I have to distribute them among my 9 friends. How can I do it? Dividing 537 by 9 and I have left with some chocolates(remainder) that means 537 is not divisible by 9 exactly. Dividing is simple to check that the number is exactly divided by the divisor ie Remainder is 0 or not when we have 2 or 3-digit numbers. If the number is too large and it takes a long time to perform the actual division. How can we know that a number is divisible by a particular divisor or not. Here comes the concept of divisibility rules: quick easy and simplest way to find out the divisibility of a number by a particular divisor.
Divisibility Rule For 2
A number is divisible by 2 if the last digit of the number is any of the following digits 0, 2, 4, 6, 8. The numbers with the last digits 0, 2, 4, 6, 8 are called even numbers, Eg: 2580, 4564, 90032 etc. which are divisible by 2.
Divisibility Rules For 3 And 9
A number is divisible by 3 if the sum of its digits is divisible by 3. Eg: 90453 (9 + 0 + 4 +5 + 3 = 21) 21 is divisible by 3. 21 = 3 × 7. Therefore, 90453 is also divisible by 3. Same rule is also applicable to test whether the number is divisible by 9 or not but the sum of the digits of the number should divisible by 9 in the above example 90453 when we add the digits we get the result as 21 which is not divisible by 9. Eg: 909, 5085, 8199, 9369 etc are divisible by 9. Consider 909 (9 + 0 + 9 = 18). 18 is divisible by 9(18 = 9 × 2). Therefore, 909 is also divisible by 9.
A number which is divisible by 9 also divisible by 3 but a number that is divisible by 3 not have surety that it is divisible by 9.
Eg: 18 is divisible by both 3 and 9 but 51 is divisible only by 3, can’t be divisible by 9.
Divisibility Rules For 5 And10
A number is divisible by 5 if that last digit of that number is either 0 or 5. Eg:500985
3456780, 9005643210, 12345678905 etc.
A number is divisible by 10 if it has only 0 as its last digit. Eg: 89540, 3456780, 934260 etc. A number which is divisible by 10 is divisible by 5 but a number which is divisible by 5 may or may not be divisible by 10.10 is divisible by both 5 and 10 but 55 is divisible only by 5 not by 10.
Divisibility Rules For 4 , 6 And 8
A number is divisible by 6 if it is divisible by both 2 and 3.
A number is divisible by 4 if the last two digits are divisible by 4. Eg: 456832960 In this example the last two digits are 60 that are divisible by 4 i.e.15 × 4 = 60. Therefore, the total number is divisible by 4.
Considering the same example let us check the divisibility rule for 8. If a number is divisible by 8 its last three digits should be divisible by 8 i.e. 960 which is divisible by 8 therefore the total number is divisible by 8.
Divisibility Rule For 11 And7
Consider the above number that we used to test the divisibility with 4 and 8
456832960 mark the even place values and odd place values. Sum up the digits in even place values together and sum up the digits in odd place values together.
| Digits | Place Value |
| 4 | 0 |
| 5 | 1 |
| 6 | 2 |
| 8 | 3 |
| 3 | 4 |
| 2 | 5 |
| 9 | 6 |
| 6 | 7 |
| 0 | 8 |
Now we have to sum up the digits in even place values ie 0 + 2 + 4 + 6 + 8 = 4 + 6 + 3 + 9 + 0 = 22
Now we have to sum up the digits in odd place values ie
1+ 3 + 5 + 7 = 5 + 8 + 2 + 6 = 21
Now calculate the difference between the sum of digits in even place values and the sum of digits in odd place values if the difference is divisible by 11 the complete number ie 456832960 is divisible by 11
Here the difference is 1, (22-21) it is divisible by 11. Therefore, 456832960 is divisible by 11.
Consider the number 5497555 to test if it is divisible by 7 or not.
Add the last two digits to the twice of the remaining number repeat the same process until it reduces to a two-digit number if the result obtained is divisible by 7 the number is divisible by 7.
55 + 2(54975) = 109950 + 55 = 110005
05 + 2(1100) = 2200 + 05 = 2205
05 + 2(22) = 44 + 5 = 49
Reduced to two-digit number 49 which is divisible by 7 i.e 49 = 7 × 7
Therefore, 5497555 is divisible by 7.
Some More Divisibility Rules
Co-primes are the pair of numbers that have 1 as the common factor. If the number is divisible by such co-primes the number is also divisible by product of the co-primes. Eg: 80 is divisible by both 4 and 5 they are co-primes have only 1 as the common factor so the number is also divisible by 20 the product of 4 and 5
21 = 3 × 7
12 = 3 × 4
22 =11 × 2
14 = 2 × 7
15 = 3 × 5
30 = 3 × 10
18 = 2 × 9
28 = 4 × 7
26 =13 × 2.
If a number is divisible by some numbers say X that number is also divisible by factors of x. Eg: if a number is divisible by 40 then it is divisible by its factors Ie: 5, 10, 2, 4, 8, 20.
Divisibility Rule For 13
If a number to be divisible by 13 add 4 times the last digit of the number to the rest of the number repeat this process until the number becomes two digits if the result is divisible by 13 then the original number is divisible by 13. Eg: 333957
(4 × 7) + 33395 = 33423
(4 × 3) + 3342 = 3354
(4 × 4) + 335 = 351
(1 × 4) + 35 = 39
Reduced to two-digit number 39 is divisible by 13 Therefore 33957 is divisible by 13.
Similar Reads
Number Theory in Mathematics
Number theory is a branch of mathematics that studies numbers, particularly whole numbers, and their properties and relationships. It explores patterns, structures, and the behaviors of numbers in different situations. Number theory deals with the following key concepts: Prime Numbers: Properties, d
4 min read
Number System
What are Numbers?
Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world. Numbers are like tools that help us: Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show
15+ min read
Number System in Maths
Number System is a method of representing numbers on the number line with the help of a set of Symbols and rules. These symbols range from 0-9 and are termed as digits. Let's learn about the number system in detail, including its types, and conversion. Number System in MathsNumber system in Maths is
13 min read
Real Numbers
Real Numbers are continuous quantities that can represent a distance along a line, as Real numbers include both rational and irrational numbers. Rational numbers occupy the points at some finite distance and irrational numbers fill the gap between them, making them together to complete the real line
10 min read
Operations on Real Numbers
Real Numbers are those numbers that are a combination of rational numbers and irrational numbers in the number system of maths. Real Number Operations include all the arithmetic operations like addition, subtraction, multiplication, etc. that can be performed on these numbers. Besides, imaginary num
9 min read
Rational Numbers
Rational numbers are a fundamental concept in mathematics, defined as numbers that can be expressed as the ratio of two integers, where the denominator is not zero. Represented in the form p/q​ (with p and q being integers), rational numbers include fractions, whole numbers, and terminating or repea
15+ min read
Decimal Expansions of Rational Numbers
Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. So in this article let's discuss some rational and irrational numbers an
6 min read
Representation of Rational Numbers on the Number Line | Class 8 Maths
Rational numbers are the integers p and q expressed in the form of p/q where q>0. Rational numbers can be positive, negative or even zero. Rational numbers can be depicted on the number line. The centre of the number line is called Origin (O). Positive rational numbers are illustrated on the righ
5 min read
Rational Numbers Between Two Rational Numbers | Class 8 Maths
Real numbers are categorized into rational and irrational numbers respectively. Given two integers p and q, a rational number is of the form p/q, where q > 0. A special case arises when q=1 and the rational number simply becomes an integer. Hence, all integers are rational numbers, equal to p. Th
6 min read
Decimal Expansion of Real Numbers
The combination of a set of rational and irrational numbers is called real numbers. All the real numbers can be expressed on the number line. The numbers other than real numbers that cannot be represented on the number line are called imaginary numbers (unreal numbers). They are used to represent co
6 min read
Laws of Exponents for Real Numbers
Laws of Exponents are fundamental rules used in mathematics to simplify expressions involving exponents. These laws help in solving arithmetic problems efficiently by defining operations like multiplication, division, and more on exponents. In this article, we will discuss the laws of exponent for r
6 min read
Irrational Numbers- Definition, Examples, Symbol, Properties
Irrational numbers are real numbers that cannot be expressed as fractions. Irrational Numbers can not be expressed in the form of p/q, where p and q are integers and q ≠0. They are non-recurring, non-terminating, and non-repeating decimals. Irrational numbers are real numbers but are different from
12 min read
Rationalization of Denominators
Rationalization of Denomintors is a method where we change the fraction with an irrational denominator into a fraction with a rational denominator. If there is an irrational or radical in the denominator the definition of rational number ceases to exist as we can't divide anything into irrational pa
8 min read
Roots
N-th root of a number
Given two numbers n and m, find the n-th root of m. In mathematics, the n-th root of a number m is a real number that, when raised to the power of n, gives m. If no such real number exists, return -1. Examples: Input: n = 2, m = 9Output: 3Explanation: 32 = 9 Input: n = 3, m = 9Output: -1Explanation:
10 min read
Squares and Square Roots
Squares and Square roots are highly used mathematical concepts which are used for various purposes. Squares are numbers produced by multiplying a number by itself. Conversely, the square root of a number is the value that, when multiplied by itself, results in the original number. Thus, squaring and
13 min read
How to Find Square Root of a Number?
In everyday situations, the challenge of calculating the square root of a number is faced. What if one doesn't have access to a calculator or any other gadget? It can be done with old-fashioned paper and pencil in a long-division style. Yes, there are a variety of ways to do so. Let's start with dis
13 min read
Cube Roots
A cube root of a number a is a value b such that when multiplied by itself three times (i.e., [Tex]b \times b \times b[/Tex]), it equals a. Mathematically, this is expressed as: b3 = a, where a is the cube of b. Cube is a number that we get after multiplying a number 3 times by itself. For example,
7 min read
Ratio, Fractions, and Percentages