The number system is one of the most important topics for competitive exams.
In this article, I have given some notes on the Number System which is very helpful from an exam point of view.
Read this article up to the end, this article will help you to clear your concept on this topic.
Let's start the topic: Number System
Table of Content |
What is a number? A number is a mathematical value used to count, measure, or label objects. Numbers are used to perform arithmetic calculations.
"A number system is defined as a system of writing or expressing numbers."
The number system allows us to understand, classify, and perform operations with numbers.
Problems on Train Questions with Solution
The number system can be categorized into the following types-
Natural Number: These are all the positive integers starting from 1 till infinity. Natural numbers are also used for counting purposes.
Examples 1, 2, 3, 4, and so on.
Note: 0 is not a natural number.
Whole Number: Whole number is similar to Natural Number but it includes 0. We can say, all natural numbers are whole numbers, but all whole numbers are not natural numbers.
Example:
Integer: An integer is a whole number (not a fractional or decimal number) that can be positive, negative, or zero.
Example: ….-2, -3, -4 0, 2, 3, 4, 6…
Rational Number: A rational number is a number that can be written as a fraction, where both the numerator and the denominator are integers, and the denominator is not equal to zero.
Example: 1/2, 3/5, 2/5, 1/7… etc.
Irrational Number: An irrational number is a real number that cannot be expressed as a fraction of two integers. The decimal expansion of an irrational number is neither terminating nor recurring.
Example: pi(π=3⋅14159265…), √2, √3,
Real Number: Real numbers are numbers that include both rational and irrational numbers. Rational numbers such as integers (-2, 0, 1), fractions(1/2, 2.5), and irrational numbers such as √3, π(22/7), etc.
Imaginary Number: Imaginary numbers are those numbers that we can just imagine but cannot physically perceive. Imaginary numbers are defined as the square root of negative numbers, which do not have a definite value.
Example √-1, √-3, etc, √-1, etc.
Even Number: An even number is a number that can be divided by 2 without a remainder. The last digit of an even number is either 0, 2, 4, 6, or 8.
Example: 2, 32, 78, 54… etc.
Odd Number: An odd number is a number that cannot be divided by 2 evenly. When divided by 2, an odd number will leave a remainder.
Example: 1, 3, 5, 25, 37… etc.
Prime Number: A prime number is a natural number greater than 1 that has only two factors, 1 and the number itself.
Example: 2, 3, 5, 7, 11, 13…… etc.
Step 1 Find the nearest square root number of the given number. Let a be the given number and n be the smallest natural number where n^{2}>a.
Step:2 Then divide this number a by n and every prime number smaller than n.
Step 3 If a number a is not exactly divisible by any number then it is a prime number.
Example: 1. Check that the number 173 is a prime number.
Solution:
The nearest square root number is 196. 196 is the square root of 14.
Then-
14^{2}>173
Divide 173 by all prime numbers less than 14 (All prime numbers less than 14 are = 2, 3, 5, 7, 11, 13).
173 is not divisible by any of these numbers so it is clear that 173 is a prime number.
Example:2. Check if 437 is a prime number or not.
The nearest square root number is 441. 441 is the square root of 21.
Then-
21^{2}>437
Divide 437 by all prime numbers less than 21 (All prime numbers less than 21 are = 2, 3, 5, 7, 11, 13, 17, 19).
437 is divisible by 19 so it is clear that 437 is not a prime number.
Hope it is clear for you.
Co-Prime Number: Co-prime numbers are those numbers that have only one common factor, ie. 1. Common factors are those natural numbers that commonly appear as a factor of two or more numbers. That means a pair of numbers is said to be co-prime when they have their highest common factor as 1.
Example: (2, 3), (3, 5), (4, 5), (7,11)... etc.
Composite Number: A number that is divisible by a number apart from 1 and the number itself, is called a composite number. A composite number is a positive integer that can be formed by multiplying two smaller positive integers.
Example: 12 (the product of the two smaller integers: 2 and 6), 4, 6, 9, 21… etc.
Complex Number: A complex number is the sum of a real number and an imaginary number.
A complex number is of the form a + ib and is usually represented by z.
Here both a and b are real numbers. The value 'a' is called the real part which is denoted by Re(z), and 'b' is called the imaginary part which is denoted by Im(z). Also, it is called an imaginary number. i is the unit imaginary number = √−1
Example: 2+i, −2 + πi, 0.4 − 2.1i… etc.
Face Value: Face value is the actual value of a digit.
Example: The face of 3 in 74632 is 3.
Place Value: Place value is the position of a digit in a number. to get the place value of the numbers, we need to multiply each number by the digit value.
Example: Find the place value of each digit in 24687.
7 is in one place so the place value of 7 = 7 x 1 = 7
8 in tens place so the place value of 8 = 8 x 10 = 80
6 in hundreds so the place value of 6 = 6 x 100 = 600
4 in thousand places so the place value of 4 = 4 x 1000 = 4000
2 in ten thousand places so the place value of 2 = 2 x 10000 = 20000
Hence 24687 = 20000+4000+600+80+7
Place value for decimal number: The first digit after the decimal represents the tenth place. The second place digit after the decimal represents the hundredth place. The third place digit after the decimal represents the thousand place and so on.
Example: Find the place value of each digit in 74.893.
3 is in thousands of places after decimal so the place value of 3 = 3 x 1/1000 = 0.003
9 is in the hundreds place after decimal so the place value of 9 = 9 x 1/100 = 0.09
8 is in tens place after decimal so the place value of 8 = 8 x 1/10 = 0.8
4 is in one place so the place value of 4 = 4 x 1 = 4
7 is in tens place so the place value of 7 = 7 x 10 = 70
Hence 74.893 = 70+4+0.8+0.09+0.003
What is the divisibility rule and why do we use it? Divisibility rules are introduced as a shortcut to find out if an integer is divisible by a number without doing the entire division process.
Here are some divisibility rules:
Divisibility rule of 1
Every number is divisible by 1. Any number divided by 1, gives the number itself.
Divisibility rule of 2
If the number is even or the last digit of any number is 0, 2, 4, 6, or 8 then the number will be completely divisible by 2.
Example: 486 is divisible by 2 because the last digit 6 is divisible by 2.
Divisibility rule of 3
If the sum of the digit of any number is divisible by 3 then the number will also be divisible by 3.
Example: 258 is divisible by 3 because the sum of 258 = 2+5+8 = 15, is divisible by 3.
Divisibility rule of 4
If the two last digits of any number is divisible by 4 then the number will also be divisible by 4.
Example: 2384 is divisible by 4 because the last two digits i.e. 84, is divisible by 4.
Divisibility rule of 5
If the unit digit of a number is 0 and 5 then the number will be completely divisible by 5.
Example: 1000, 550, 425, 545… etc
Divisibility rule of 6
If the number is divisible by both 2 and 3 then the number will also be divisible by 6.
Example: 924 is divisible by both 2 and 3 because the last digit 4 is divisible by 2 and the sum of the digit 9+2+4 = 15, is divisible by 3. So the number 924 is divisible by 6.
Divisibility rule of 7
If the difference between twice the unit digit and the remaining part of the number is a multiple of 7 or equal to 0 then the number will be divisible by 7.
Example: 224 is divisible by 7 or not?
The unit digit of the number is 4. Step 1: twice the unit digit of the number = 4*2 = 8 Remaining part = 22 Step 2: subtract the twice of the unit digit from the remaining part = 22-8 = 14 So the difference between the two numbers is 14 which is divisible by 7. Hence the number 224 is divisible by 7. |
Divisibility rule of 8
If the last three digits of any number is divisible by 8 then the number will also be divisible by 8.
Example: 725416 is divisible by 8 because the last three digits i.e. 416 are divisible by 8.
Divisibility rule of 9
If the sum of the digit of any number is divisible by 9 then the number will also be divisible by 9.
Example: 3759696 is divisible by 9 because the sum of 375966 = 3+7+5+6+9+6 = 36 is divisible by 9.
Divisibility rule of 10
If the last digit of any number is 0, then the number will be divisible by 10.
Example: 10, 120, 80, 20 etc.
Divisibility rule of 11
If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11 completely.
Sum of digits in odd places – Sum of digits in even places = 0 or a multiple of 11
Example: 82907 is divisible by 11?
82907 = 8+9+7 - 2+0 = 24 - 2 = 22
22 is divisible by 11 then the number 82907, will also be divisible by 11.
Divisibility rule of 12
If the number is divisible by both 3 and 4, then the number will also be divisible by 12.
Example: 2052 is divisible by 12?
Some of the digit 2052 = 2+0+5+2 = 9, divisible by 3
The last digit of the number is 52 which is divisible by 4.
So the number is divisible by both 3 and 4, then the number also be divisible by 12.
If on dividing a number A by a number B the quotient is Q and the remainder is R, then the formula is used -
Dividend = Divisor x Quotient + Remainder |
Dividend = A
Divisor = B
Quotient = Q
Remainder = R
Then according to the formula -
A = B x Q +R
Type 1:
The quotient in a division sum is 403. The divisor is 100 and the remainder is 58. The dividend is ……… ?
Solution:
Quotient = 403
Divisor = 100
Remainder = 58
Dividend =?
Formula = Dividend = Divisor x Quotient + Remainder
Dividend = 100 x 403 + 58
Dividend = 40300+58
Dividend = 40358
Type 2:
A number when divided by 136 leaves the remainder 36. If the same number is divided by 17, the remainder will be -
1st Method:
Divisor = 136
Remainder = 36
Let the Dividend = a and Quotient = b
Formula = Dividend = Divisor x Quotient + Remainder
a = 136 x b + 36
136 is a multiple of 17, so we can write it as follows - |
a = (17 x 8b) + 36
a = 17 x 8b + (17 x 2) +2 [Note: Try to make an equation in multiple of 17, so we can write 36 = 17 x 2 +2, i.e. 34+2 ]
a = 17 x 8b + 17 x 2 +2
a = 17 x (8b +2) + 2
So -
Dividend = a
Divisor = 17
Quotient = 8b +2
Remainder = 2
Hence if the same number is divided by 17, the remainder will be 2.
2nd Method:
Divisor = 136
Remainder = 36
Let the Dividend = a and Quotient = 1
Formula = Dividend = Divisor x Quotient + Remainder
a = 136 x 1 + 36
a = 136+36
a = 172
Now -
172 is divided by 17.
172/2
= 170+2/17
Then the remainder is 2.
Type 3:
What is the smallest number that should be subtracted from 4000 so that the resulting number is completely divisible by 19?
Solution:
On dividing 4000 by 19 we get the remainder = 10
When we subtract 10 from 4000, we get 3990 i.e. is completely divisible by 19.
Type 4:
The smallest number that must be added to 803642 to obtain a multiple of 11 is
1st Method:
On dividing 803642 by 11 we get remainder = 4
Here we will not add 4 to 803642 because if we add 4 to 803642 then we get = 803646
Which is not divisible by 11.
So the required number = 11-4 = 7
When we add 7 to 803642, we get = 803649
Which is divisible by 11.
2nd Method
Using divisibility rule of 11.
If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11 completely.
Sum of digits in odd places – Sum of digits in even places = 0 or a multiple of 11
The sum of a digit in odd place = 0+6+2 = 8
The sum of digits in an even place = 8+3+4 = 15
Difference = 15-8 = 7
So the required number = 7
Type 5:
The smallest 3-digit number that is completely divisible by 14
Solution:
The smallest 3-digit number = 100
On dividing 100 by 14 we get the remainder = 2
The required number to add = 14 - 2 = 12
So the smallest 3 digit number that is completely divisible by 14 = 100+12 = 112
Note: Here we will not subtract the remainder from 100, if we do that, the number will not be 3 digit number.
Type: 6
Find the greatest 5-digit number which is completely divisible by 463.
The greatest 5-digit number = 99999
On dividing 99999 by 463 we get the remainder = 454
So the required number = 99999 - 454 = 99545
Note: Here we will not add the remainder in 99999 because if we do that the number will not be the 5-digit number.
Here are some important formulas that we use to solve number system questions.
The Sum of n natural number = n(n+1)/2
The Sum of the square of n natural number = [n(n+1)(2n+1)]/6
The Sum of the cube of n natural number = [n(n+1)/2]^{2}
The Sum of the first n odd numbers = n^{2}
The Sum of first n even number = n(n+1)
Example:
Find the sum of all natural numbers from 1 to 100.
1+2+3+4+..........+100
Solution:
Formula: n(n+1)/2
n=100
Sum = 100(100+1)/2
=50*51
= 2550
Find the sum of all even numbers from 1 to 20.
2+4+8+10+ ….+20
Solution:
Formula: n(n+1)
n=10
Sum= 10(10+1)
= 10*11
= 110
Find the sum of all odd numbers from 1 to 20.
1+3+5+......+19
Solution:
Formula: n^{2}
n= 10
Sum = 10^{2}
= 100
Find the sum of the squares from 1 to 9.
1^2+2^2+3^2……+10^2
Solution:
Formula =n(n+1)(2n+1)/6
n=9
Sum = 9(9+1)(2*9+1)/6
= 9*10(18+1)/6
= 90*19/6
= 15*19
= 285
Find the sum of the cubes from 1 to 10.
1^3+2^3+3^3……+10^3
Solution:
Formula= [n(n+1)/2]^{2}
Sum = [10(10+1)/2]^2
= (10*11/2)^2
= (110/2)^2
= 55^2
= 3025
Here are some important mathematical formulas for the number system.
The general form of an Arithmetic Progression is a, a + d, a + 2d, a + 3d, and so on. Where a = first term, d = common difference = Tn - (Tn-1), and Tn = nth term.
When three quantities are in AP, the middle one is called the arithmetic mean of the other two. If a, b, and c are three terms in AP then
The general form of a Geometric Progression is a, ar, ar^{2}, ar^{3}, ar^{4}, and so on.
Where a = first term, r = common ratio = t_{n} / t_{n-1}
If a, b, and c are three quantities in GP and b is the geometric mean of a and c then b =√ac.
Units place digit is the digit which is placed at one’s place of a particular number.
For example, the unit digit for the number 458 is 8.
How to find unit digits when the numbers are multiplied?
To find the ones digit of a given number, multiply the ones digits of the given number.
Example: 476 x 198 x 359 x 242
Now multiply the ones digits of the given numbers = 6 x 8 x 9 x 2 = 864
Hence the unit digit for the given number is = 4
Finding the unit digit in a number with an exponent.
The concept of finding the unit digit in a number with an exponent is different for the numbers from 0 to 9.
When the unit digit is 0,1,5,6:
If we have to find the unit's digit in such a number in which 0, 1, 5, and 6 are present in the one’s place, then the unit's digit does not change in such numbers. That is, the unit digit remains 0,1,5,6 respectively.
Example:
The unit digit in 100^{25 }= 0 The unit digit in 123641^{421 }= 1 The unit digit in 1235^{3685 }= 5 The unit digit in 28946^{3725}= 6 |
In the above examples, the unit digit is the same as the number given in the one’s place.
When the unit digit is 2, 3, 7, 8:
2, 3, 7, and 8 have a cyclicity of 4 for the unit digits.
2^{1 }= 2,
2^{2 }= 4,
2^{3} = 8
2^{4 }= 16 = 6
and after that, the cycle will repeat.
So, the cyclicity of 2 is 4 = 2, 4, 8, 6.
3^{1 }= 3,
3^{2} = 9,
3^{3 }= 27 = 7
3^{4} = 81 = 1
and after that, the cycle will repeat.
So, the cyclicity of 3 is 4 = 3, 9, 7, 1.
7 and 8 follow similar logic.
So 2, 3, 7, and 8 have a unit digit cyclicity of 4.
So to find the unit digit for numbers 2, 3, 7, and 8, we divide the power by 4 and solve by placing the remainder in the place of the power.
Example: 1
Find the unit digit in 562^{1997}
Step 1: divide the power 1997 by 4. We get the remainder = 1.
Step 2: we will put 1 as the power in place of 1997.
Now the digit one’s place of the given number = 2
Step 3: 21 = 2
So the unit digit in 562^{1997} = 2.
Note: Dividing a number by 4 gives the same remainder as is obtained by dividing the last two digits of that number. Just as dividing 1997 by 4 gives a remainder of 1, similarly dividing the last two digits of 1997, 97, by 4 also gives a remainder of 1. Therefore, we can solve the question by dividing only the last two numbers instead of dividing the whole number. |
Example: 2
Find the unit digit in 1727^{654}
Step 1: divide the last digit of 654 i.e. 54 by 4. We get the remainder = 2
Step 2: we will put 2 as the power in place of 654.
Now the digit in the one’s place of the given number = 7
Step 3: 72 = 49 = 9
So the unit digit in 1727^{654} = 9
When the unit digit is 4
4 has the cyclicity of 2 different numbers.
4^{1} = 4
4^{2} = 16 = 6 (after it the cycle will repeat)
4^{3} = 64 = 4
Hence the cyclicity of 4 has only two numbers. If the power of 4 is an even number then the unit digit is 6 and when the power of 4 is an odd number the unit digit is 4.
Example:
Find the unit digit in (874)^{154879}.
The power is an odd number so the unit digit in (874)^{154879 }= 4
Find the unit digit in154^{45836}
The power is an even number so the unit digit in 154^{45836} = 6
When the unit digit is 9
As 4, and 9 has also the cyclicity of 2 different numbers.
9^{1 }= 9
9^{2} = 81 = 1(after it the cycle will repeat)
9^{3 }= 729 = 9
Hence the cyclicity of 9 has only two numbers. If the power of 9 is an even number then the unit digit is 1 and when the power of 9 is an odd number the unit digit is 9.
Example:
Find the unit digit in(759)^{789635}
The power of 9 is an odd number so the unit digit in (759)^{789635} = 9
Find the unit digit in (429)^{48796}
The power of 9 is an even number so the unit digit in (429)^{48796} = 1
Note: If a number is divided by 4 and the remainder is 0, then -
The unit digit for numbers 2 and 8 = 6,
The unit digit for numbers 3 and 7 = 1
(If there is 00 at the end of a number, the number will be completely divisible by 4.)
Example:
22^{98} - 53^{80}
(If we divide 98 and 80 by 4 we get the remainder 0. Here the the digits in one’s place are 2 and 3. By the above rule if the remainder is 0 and the number in the one’s place is 2 and 3 then the unit digit will be 6 and 1 respectively.)
Unit digit in 22^{98} = 6 and the unit digit in 53^{80}= 1
Then -
When the unit digit is 2, 3, 7, 8:
22^{98} - 53^{80}
= 6 - 1
= 5
So the unit digit in 22^{98} - 53^{80} = 5
Note: If the first number in subtraction is smaller than the second number then 10 will be added to the first number. That means the answer cannot be negative.
Example:
756^{145}- 149^{149}
= 6145-9149
= (unit digit is not change for 6 and the power of 9 is an odd number so the unit digit will be 9)
= 6 - 9 (her 6 is smaller than 9 so we will add 10 to 6)
= 10+6 - 9
= 16 - 9
= 7
So the unit digit in 756^{145}- 149^{149} = 7
Note: If we want to get the unit digit by multiplying numbers and somehow 2 or 5 comes in between the numbers, then on multiplying them we get zero in the end. That means the unit digit is 0. Therefore, after multiplying the numbers, the unit digit will be 0. |
Example:
What will be the unit digit when all the numbers from 1 to 50 are multiplied?
1 x 2 x 3 x 4 x 5 x 6……….x 50
(here is 2 and 5 between the numbers)
On multiplying 2 and 5 we get 10 where the unit digit is 0.
So when 0 is multiplied with other numbers we will get 0.
Hence the unit digit in 1 x 2 x 3 x 4 x 5 x 6……….x 50 = 0
The distributive law states that multiplying a number by a set of numbers added together is the same as taking each multiplication separately.
Example 1 : 4 x (2+3) = 4 x 2 + 4 x 3
Some other Examples
Find the product using distributive law.
240 x 105
Solution:
=240 x (100+5)
= 240 x 100 + 240 x 5
= 24000 + 1200
= 25200
8765 x 974 - 8765 x 874
Solution:
= 8765 x (974-874)
= 8765 x 100
= 876500
1502 x 1502
Solution
= (1502)^{2}
= (1500+2)^{2}
Formula to solve this type of questions = (a+b)^{2 }= a^{2}+b^{2}+2ab
= 1500^{2} + 2^{2}+ 2x1500x2
= 2250000 + 4 + 6000
= 2256004
1995 x 1995
Solution
= (1995)^{2}
= (2000-5)^{2}
Formula to solve this type of questions = (a+b)^{2} = a^{2}+b^{2}-2ab
= 2000^{2} + 5^{2 }- 2x2000x5
= 4000000 + 25 - 20000
= 3980000 + 25
= 3980025
883 x 883 - 117 x 117
Solution
= 883^{2} - 117^{2}
Formula to solve this type of questions = a^{2}-b^{2} = (a+b) (a-b)
= (883+117) (883-117)
= 1000 x 766
= 766000
This article is for clearing your basic concept of the number system. Learn the basic concept of the number system carefully.
In the next article, we have some important questions related to the number system.