Numbers
Important Formulae: 1) Geometric Progression: x, xr, xr
3, xr
4, --- are said to be in geometric progression. Here, a is first term and r is common ratio.
a) n
th term = xr
(n -1) | b) Sum of n terms = | x (1 – rn) | , here r < 1 |
| (1 – r) |
| c) Sum of n terms = | x (rn – 1) | , here r > 1 |
| (r – 1) |
2) Arithmetic Progression: x, x + y, x + 2y, x + 3y are said to be in A.P. Here x is first term and common difference is y.
a) n
th term = x + (n – 1) y
| b) Sum of n terms = | n | [2x + (n – 1)y] |
| 2 |
| 3) 1 + 2 + 3 ± - - - - + n = | n(n + 1) |
| 2 |
| 4) (12 + 22 + 32 - - - - + n2) = | n(n + 1) (2n + 1) |
| 6 |
| 5) (13 + 23 + 33 - - - - + n3) = |  | n(n + 1) |  | 2 |
| 2 |
Points to Remember: Difference between Arithmetic Progression and Geometric Progression: Arithmetic Progression: It is the sequence of numbers in which each term after first is obtained by adding a constant to preceding term. The constant term is called as the common difference.
Geometric Progression: It is a sequence of non-zero numbers. The ratio of any term and its preceding term is always constant.
| Types of Numbers | Definition | Example | Points to remember |
| Natural Numbers | Numbers used for counting and ordering | 1, 2, 3, 4, 5, ----- natural numbers | |
| Whole Numbers | All counting numbers along with zero form a set of whole numbers | 0, 1, 2, 3, 4 ------ whole numbers | Any natural number is a whole number
0 is a whole no. which is not a natural no. |
| Integers | Counting numbers + negative counting numbers + zero, all are integers | -2, -1, 0, 1, 2, ---- integers | Positive integers: 0, 1, 2, 3, -------
Negative integers: -1, -2, -3, -4, --------- |
| Even Numbers | Number divisible by 2 is called as even number | 0, 2, 4, 6, 8, ------ even numbers | |
| Odd Numbers | Number not divisible by 2 is called as even number | 1, 3, 5, 7, 9, ------ odd numbers | |
| Prime Numbers | A number having exactly two factors i.e 1 and itself is called as prime number | 2, 3, 5, 7, 11, ----- prime numbers | |
| Composite Numbers | Natural numbers which are not prime numbers are called as composite numbers | 4, 6, 8, 9, 10, ----- composite nos. | |
| Co Primes | Any two natural numbers x and y are co-prime if their HCF is 1 | (4, 5), (7, 9), ---Co-prime numbers | |
Divisibility of Numbers1) Number divisible by 2Units digit – 0, 2, 4, 6, 8
Ex: 42, 66, 98, 1124
2) Number divisible by 3Sum of digits is divisible by 3
Ex: 267 ---(2 + 6 + 7) = 15
15 is divisible by 3
3) Number divisible by 4 Number formed by the last two digits is divisible by 4
EX: 832 The last two digits is divisible by 4, hence 832 is divisible by 4
4) Number divisible by 5Units digit is either zero or five
Ex: 50, 20, 55, 65, etc
5) Number divisible by 6The number is divisible by both 2 and 3
EX: 168 Last digit = 8 ---- (8 is divisible by 2)
Sum of digits = (1 + 6 + 8) = 15 ----- (divisible by 3)
Hence, 168 is divisible by 6
6) Number divisible by 11If the difference between the sums of the digits at even places and the sum of digits at odd places is either 0 or divisible by 11.
Ex: 4527039Digits on even places: 4 + 2 + 0 + 9 =15
Digits on odd places: 5 + 7 + 3 = 15
Difference between odd and even = 0
Therefore, number is divisible by 11
7) Number divisible by 12 The number is divisible by both 4 and 3
Ex: 1932 Last two digits divisible by 4
Sum of digits = (1 + 9 + 3 + 2) = 15 ---- (Divisible by 3)
Hence, the number 1932 is divisible by 12
Basic Formulae: (Must Remember) 1) (a - b)
2 = (a
2 + b
2 - 2ab)
2) (a + b)
2 = (a
2 + b
2 + 2ab)
3) (a + b) (a – b) = (a
2 – b
2 )
4) (a
3 + b
3) = (a + b) (a
2 – ab + b
2)
5) (a
3 - b
3) = (a - b) (a
2 – ab + b
2)
6) (a + b + c)
2 = a
2 + b
2 + c
2 + 2 (ab + bc + ca)
7) (a
3 + b
3 + c
3 – 3abc) = (a + b + c) (a
2 + b
2 + c
2 – ab – bc – ac)
Quick Tips and Tricks: 1) If H.C.F of two numbers is 1, then the numbers are said to be co-prime.
To find a number, say b is divisible by a, find two numbers m and n, such that m*n = a, where m and n are co-prime numbers and if b is divisible by both m and n then it is divisible by a.
2) Sum of the first n odd numbers = n
23) Sum of first n even numbers = n ( n + 1)
4) Even numbers divisible by 2 can be expressed as 2n, n is an integer other than zero.
5) Odd numbers which are not divisible by 2 can be expressed as 2(n + 1), n is an integer.
6) Dividend = [(Divisor × Quotient)] + Remainder
7) If Dividend = a
n + b
n or a
n – b
na) If n is even: a
n - b
n is divisible by (a + b)
b) If n is odd: a
n + b
n is divisible by (a + b)
c) a
n - b
n is always divisible by (a – b)
8) To find the unit digit of number which is in the form a
b. (Ex: 7
105, 9
125)
1) If b is not divisible by 4Step 1: Divide b by 4, if it is not divisible then find the remainder of b when divided by 4.
Step 2: Units digit = ar, r is the remainder.
2) If b is multiple of 4Units digit is 6: When even numbers 2, 4, 6, 8 are raised to multiple of 4.
Units digit is 1: When odd numbers 3, 7 and 9 are raised to multiple of 4.
Question VarietyGenerally 6 types of numerical are asked from this chapter. Understanding and studying the concepts will help in solving the numerical related to this chapter. Type 1: Find units digit of a number in the form of ab
Find the unit digit of (4137)
754 a. 9
b. 7
c. 3
d. 1
View solution Correct Option:(a)
Hint: Divide b by 4, if it is not divisible then find the remainder of b when divided by 4.
Units digit = ar, r is the remainder
Number is in the form ab i.e (4137) 754
4 × 188 = 752, therefore we get remainder as 2
Units digit = (4137)2 = 17114769
9 is the digit in units place
Find the unit digit in the product (3
65 × 6
59 × 7
71)
a. 1
b. 4
c. 5
d. 9
View solution Correct Option: (b)
Hint:
If b is multiple of 4
Units digit is 6 : When even numbers 2, 4, 6, 8 are raised to multiple of 4.
Units digit is 1 : When odd numbers 3, 7 and 9 are raised to multiple of 4.
Using the hint given, we can easily solve product of large numbers.
[3(4)16 × 3] = (1 × 3) = 3
[659] = 6
[771] = [7 (4)17 × 73] = [1 × 3] = 3
Therefore, (3 × 6 × 3) = 54
Required unit digit is 4.
