## 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 – r^{n}) | , here r < 1 |

(1 – r) |

c) Sum of n terms = | x (r^{n} – 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) (1^{2} + 2^{2} + 3^{2} - - - - + n^{2}) = | n(n + 1) (2n + 1) |

6 |

5) (1^{3} + 2^{3} + 3^{3} - - - - + n^{3}) = | 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 Numbers**

**1) Number divisible by 2**

Units digit – 0, 2, 4, 6, 8

**Ex:**42, 66, 98, 1124

**2) Number divisible by 3**

Sum 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 5**

Units digit is either zero or five

**Ex:**50, 20, 55, 65, etc

**5) Number divisible by 6**

The 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 11**

If 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: 4527039**

Digits 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

^{2}

**3)**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

^{n}

a) 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 4**

**Step 1:**Divide b by 4, if it is not divisible then find the remainder of b when divided by 4.

**Step 2: Units digit = a**r is the remainder.

^{r},**2) 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.

**Question Variety**

**Generally 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

**Q 1.**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 = a^{r}, r is the remainder

Number is in the form a^{b} 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

**Q 2.**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

[6^{59}] = 6

[7^{71}] = [7^{ (4)17} × 7^{3}] = [1 × 3] = 3

Therefore, (3 × 6 × 3) = 54

Required unit digit is 4.