List of Some Interesting Numbers

 Dear Friends ---

Numbers are very fascinating. Every number is unique and I have written the properties of number 1 - 100 in my blog highlighting some of their characteristics. 

I have been very interested in collecting the name of some of the numbers alphabetically and the list is yet incomplete. Please find the name of some of the numbers I have collected from different sources --

 

1.   Abundant Number

2.   Achilles Number

3.   Admirable Number

4.   Alternating Number

5.   Amenable Number

6.   Amicable Number

7.   Anti- Perfect Number

8.   Apocalyptic Number

9.   Astonishing Number

10.                Automorphic Number

11.                Beast Number

12.                Bell Number

13.                Binomial Number

14.                Cake Number

15.                Cube Number

16.                Catalan Number

17.                Cardinal Number

18.                Composite Number

19.                Congruent Number

20.                Cullen Number

21.                Cyclic Number

22.                Deficient Number

23.                Deceptive Number

24.                Decagonal Number

25.                Economical Number

26.                Esthetic Number

27.                Eulerian Number

28.                Evil Number

29.                Even number

30.                Factorial Number

31.                Fibonacci Number

32.                Friedman Number

33.                Gilda Number

34.                Happy Number

35.                Harmonic Number

36.                Harshad Number

37.                Heptagonal number

38.                Hex number

39.                Hexagonal number

40.                Highly composite number

41.                Hoax number

42.                Hungry number

43.                Impolite number

44.                Inconsummate number

45.                Irrational number

46.                Junctioin number

47.                Kaprekar number

48.                Lehmer number

49.                Leyland number

50.                Lonely number

51.                Lucas number

52.                Lucky number

53.                Magic number

54.                Modest number

55.                Motzkin number

56.                Narcissistic number

57.                Natural number

58.                Nonagonal number

59.                Nude number

60.                Octagonal number

61.                Odd number

62.                Ormiston number

63.                Palindrome number

64.                Pancake number

65.                Pandigital number

66.                Partition number

67.                Pentagonal number

68.                Perfect number

69.                Perrin number

70.                Persistent number

71.                Palindrome number

72.                Practical number

73.                Prime number

74.                Primorial number

75.                Pseudo perfect number

76.                Rare number

77.                Rational number

78.                Real number

79.                Repdigit number

80.                Sastry number

81.                Self number

82.                Semi prime number

83.                Sliding number

84.                Smith number

85.                Sophie German number

86.                Square number

87.                Taxi cab number

88.                Tetrahedral number

89.                Tetranacci number

90.                Transcendental number

91.                Triangular number

92.                Tribonacci number

93.                Twin prime number

94.                Uban number

95.                Ulam number

96.                Untouchable number

97.                Vampire number

98.                Wasteful number

99.                Weird number

100.            Woodall number

101.            Zeisel number

102.            Zuckerman number

 

I do promise that I will start writing about all these numbers in coming days but that needs your support. 

Regards 

 Dr Rajesh Kumar Thakur

 

 

FRIENDLY NUMBER

                                                                      FRIENDLY NUMBERS

FRIENDLY NUMBER is different from Friend / Amicable Numbers. So don't get confused with its similarity.

Friendly Number

Two or more natural numbers are called friendly numbers if they have common abundancy index. The common abundancy index is decided by dividing the sum of all divisors of a number including itself by the number itself.

Example:- 30 has the divisors – 1, 2, 3, 5, 6, 10, 15 and 30

The sum of divisors = 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 = 72

Abundancy index = 72/30 = 12/5

The abundancy index of a number could be a natural number or rational number.

If the sum of divisors of a number  (excluding itself) = Number itself , it is called Perfect Number

If the sum of divisors of a number  (excluding itself) >Number itself , it is called Abundant Number

If the sum of divisors of a number  (excluding itself) < Number itself , it is called Deficient Number

6 and 28 are Perfect Numbers

Divisor of 6 = 1, 2 and 3 . Sum of divisors = 1 + 2 + 3 = 6

Divisors of 28 are = 1, 2, 4 ,7 and 14 . Sum of divisors = 1 + 2 + 4 + 7 + 14 = 28

6 and 28 are friendly number with abundancy index 2, i.e. any number with abundancy index 2 is called Friendly numbers.

Divisors of 6 = 1, 2, 3 and 6

Sum of Divisors = 1 + 2 + 3 + 6 = 12

Abundancy Index = 12 /6 = 2

Divisor of 28 = 1, 2, 4, 7, 14 and 28

Sum of divisors = 1 + 2 + 4 + 7 + 14 + 28 = 56

Abundancy Index = 56 / 28 = 2

Since 6 and 28 have same abundancy index so they are FRIENDLY NUMBERS.

More examples of friendly numbers are --- (30,140) (2480, 6200 and 40640)

 If you want to know about Amicable/Friends Number 

FRIEND NUMBER/ AMICABLE NUMBERS

READ ANOTHER ARTICLE 

Numbers with Cool Names: Amicable, Sociable, Friendly – TOM ROCKS MATHS

LIST OF DIFFERENT NUMBERS

Do write your comment in comment box.

DR RAJESH KUMAR THAKUR

 

 

 

Finding Remainder when Dividend is in power

Dr Rajesh Kr Thakur

Finding Remainder Easy Part - 1

 

Finding Remainder

Do you think finding remainder is easy?

Yes, if you are dividing one big number by smaller.

But consider a situation where you find yourself perplexed and helpless and you don’t have any idea to calculate the remainder because you can’t apply the basic rule –

Dividend = Divisor × Quotient + Remainder

Let’s first begin with Euclid Division lemma

As you know that Euclid is called the father of geometry but he has contributed immensely in Arithmetic and Number theory also.

Euclid Division Lemma

Let a and b be any two positive integers then there exist unique integers q and r such that a = bq + r where 0 ≤ r < b

This is basically the Euclid Division Lemma

                          

In the first example Dividend = 445, Divisor = 17,   Quotient = 26 and Remainder = 3

Apply the division lemma

Dividend = Divisor × Quotient + Remainder

          445 = 17 × 26 + 3

                 = 442 + 3

Here, Remainder < Divisor

In the second example, remainder = 0 which also satisfies the division lemma. Let’s check

Dividend = Divisor × Quotient + Remainder

2412 = 36 × 67 + 0

2412 = 2412

But what will be the remainder when 2 is divided by 6?

Remember, if numerator is less than denominator, i.e. when dividend is less than divisor then dividend will be remainder.

Here, obviously, 2 is the remainder

Some people write 2 ÷ 6 as and further cancel out to get and write 1 as quotient which is wrong. Finding remainder is something different from simplifying fraction or changing a fraction into simplest form. Therefore, you are advised not to cancel the number.

In the above two examples we have seen the remainder is a positive integer but in finding the remainder of larger number the concept of negative integer remainder also plays a big role.

Negative Remainder

Let’s take an example

26 = 9 × 2 + 8

26 = 9 × 3 – 1

In the first case remainder = + 8 while in the second case remainder = –1.

Negative remainder reduces the effort of calculation.

54 = 11 × 4 + 10                                             Remainder = + 10

54 = 11 × 5 – 1                                                Remainder = – 1

 

Finding remainder of product

Example: - Find the remainder when 33 x 34 x 35 is divided by 8

Solution: - The first thing we can do it to multiply 33, 34 and 35

                        33 × 34 × 35 = 39270

Now divide 39270 by 8 . What do you get?

Remainder = 6

Let’s try it by remainder method.

On division of 33 by 8 we get remainder = 1

On division of 34 by 8 we get remainder = 2

On division of 35 by 8 we get remainder = 3

Hence  Remaider = 6

The conclusion that we can draw from the above example is that if a, b, c ---- be numbers divided by w gives remainder x, y, z - - respectively then the product of a, b, c - - - when divided by the same divisor w will give the same remainder obtained by dividing the product of remainder x, y, z --- by w.

First divide 121, 118 and 124 by 60 to get remainders as 1, –2 and 4 respectively.

Therefore final remainder can be obtained without actual multiplication of 121× 118 × 124 and it will be the product of individual remainder of numbers and i.e.  (1×–2×4) = –8

Final remainder = –8 + 60 = 52.

Example: Find the remainder when 37⁵⁰⁰ is divided by 9

Solution:- As we know that 3⁴ = 3 × 3 × 3 × 3 = 81

Hence, 37⁵⁰⁰ = 37 × 37 × - - - 500 times.

Multiplying 37 five hundred times is a big deal so we need to think something extra ordinary. We divide 37 by 9 and get the remainder 1

Therefore, 37⁵⁰⁰  divided by 9 is equal to 1⁵⁰⁰divided by 9

Hence, Remainder = 1

Example: Find the remainder when 54 × 63 × 86 is divided by 11

Solution:-

On division of 54 by 11 we get remainder = 10

On division of 63 by 11 we get remainder = 8

On division of 86 by 11 we get remainder = 9

Through mental calculation we can find the remainder on dividing 720 by 11.

720 = 11 × 65 + 5; Remainder = 5

Let’s apply the concept of negative integer

On division of 54 by 11 we get remainder = – 1 because 54 – 5 × 11 = – 1

On division of 63 by 11 we get remainder = – 3 because 63 – 6 × 11 = –3

On division of 86 by 11 we get remainder = – 2 because 86 – 8 × 11 = – 2

Since final remainder can’t be – 6, so we add the divisor 11 to make the remainder positive at the end.

Remainder = – 6 + 11 = 5

This shows that the negative remainder concept saves our time.

DR Rajesh Thakur