The Amazing Concepts

Narcissistic number is a number that returns the original number when each digit is raised separately to the power of the number of digits. Checking a Narcissistic Number Step 1 : Consider any number. Step 2 : (Number) power of the number of digits If the number obtained in Step 2 is same as the number considered in Step 1, it is a Narcissistic number. But if the number obtained in Step 2 is different than the number considered in Step 1, it is not a Narcissistic number . Examples Example 1 Step 1 : Consider number 0. Step 2 : 0 1 => 0. The number obtained in Step 2 is same as the number considered in Step 1. Therefore, 0 is a Narcissistic number . Example 2 Step 1 : Consider number 1. Step 2 : 1 1 => 1. The number obtained in Step 2 is same as the number considered in Step 1. Therefore, 1 is a Narcissistic number . Example 3 Step 1 : Consider number 2. Step 2 : 2 1 => 2. The number obtained in Step 2 is same as the number considered in Step 1. Therefore, 2 is a Narcissistic...

Happy number is a positive integer that gives the final number as 1 when certain steps ( square and sum the digits )   are performed continuously. Checking a Happy Number Step 1 : Consider any number. Step 2 : Square the number and replace the resultant with the sum of the squares of its digits. Step 3 : Repeat the steps. The final number will be 1, or the steps will loop endlessly in a cycle. Those numbers (considered in Step 1) that give the final number as 1 are Happy numbers . Examples Example 1 Step 1 : Consider number 1. Step 2 : Square the number => 1 2 => 1 x 1 => 1. Step 3 : Repeat the steps. 1 2 => 1 x 1 => 1 Repeating the steps gives the final number as 1. Therefore, the number considered in Step 1 is a Happy number . Example 2 Step 1 : Consider number 2. Step 2 : Square the number => 2 2 => 2 x 2 => 4. Step 3 : Repeat the steps. 4 2 => 4 x 4 => 16 1 2 + 6 2 => 1 + 36 => 37 3 2 + 7 2 => 9 + 49 => 58 5 2 + 8 => 2...

Dudeney number is a positive integer that is equal to the cube of the sum of all its digits. Checking a Dudeney Number Step 1 : Consider any number. Step 2 : Sum all its digits. Step 3 : Cube the number obtained in Step 2. If the number obtained in Step 3 is equal to the number considered in Step 1, it is a Dudeney number. But if the number obtained in Step 3 is not equal to the number considered in Step 1, it is not a Dudeney number. Examples Example 1 Step 1 : Consider number 0. Step 2 : Sum all its digits. The resultant is 0. Step 3 : Cube the number 0 obtained in Step 2 => 0 3 => 0 x 0 x 0. The resultant is 0. The number obtained in Step 3 is equal to the number considered in Step 1, hence 0 is a Dudeney number. Example 2 Step 1 : Consider number 1. Step 2 : Sum all its digits. The resultant is 1. Step 3 : Cube the number 1 obtained in Step 2 => 1 3 => 1 x 1 x 1. The resultant is 1. The number obtained in Step 3 is equal to the number considered in Step 1, hence 1 i...

Kaprekar number is one of the amazing concepts in Mathematics. The square of a number when divided into parts and none of the parts is equal to 0, and sum of both the parts forms the original number, then that number is a Kaprekar number . Let's understand this concept in detail: Checking a Kaprekar Number Following are the steps to check whether any number is a Kaprekar number or not: Step 1 : Consider any number N (must be greater than 0). Step 2 : Square the number N (N x N). Step 3 : Divide the resultant obtained in Step 2 into two parts (left and right). Possible scenarios that occur when dividing the resultant into two parts: Scenario 1 : If the resultant obtained in Step 2 is a single digit number If the resultant obtained in Step 2 is a single digit number (no matter even or odd), compare it with the number considered in Step 1.  If the number in Step 2 is equal to the number in Step 1, it is a Kaprekar number. But if the number  in Step 2 is not equal to the...

In Mathematics, one of the amazing concepts is the Reverse-Add-Then-Sort (RATS) Sequence. The RATS Sequence is a sequence that is formed by reversing , adding , then sorting the digits. Let's understand this concept in detail: Form the RATS Sequence Step 1 : Consider any number (must be greater than 0). Step 2 : Reverse the number. Step 3 : Add the original number (considered in Step 1) with the number formed by reversing the number (in Step 2). Step 4 : Sort the number obtained in Step 3 in ascending order. The resultant obtained after sorting the number forms the next sequence after n iteration. Examples First Iteration Step 1 : Number 1 . Step 2 : Reversing the number 1  gives the resultant  1 . Step 3 : Adding the original number 1 (considered in Step 1) with the number formed by reversing (in Step 2)  i.e. 1 gives the resultant 2 . Step 4 : Sorting the number obtained in Step 3 in ascending order gives the resultant 2 . Here number 2 forms the next sequence after...

You might have heard the term black hole . A black hole is a region of spacetime where gravity is so strong that even light or electromagnetic waves cannot escape it. In Mathematics, the number 123 is called the black hole number. Let’s understand this amazing concept in detail. The Procedure Step 1 : Consider any number . Step 2 : Count the number of even digits. Step 3 : Count the number of odd digits. Step 4 : Count the number of total digits. Step 5 : Consider the number (" even " " odd " " total ") obtained in the n   iteration. Step 6 : Repeat the steps ( Step 1 to Step 5 ) after every iteration until you get the number (" even " " odd " " total ") as  123 . Note : If you repeat the steps ( Step 1 to Step 5 ) even after getting the number 123, you will obtain the same number i.e. 123 after every iteration. Example First Iteration Step 1 : Number 1234567890 . Step 2 : Number of  even  digits = 5 . Step 3 : Number of...

A triangular number (triangle number) is a number that represents the pattern of dots arranged in a way that they form an equilateral triangle. An equilateral triangle (also known as a regular triangle) is the one in which all the three sides are of equal length. A triangular number is denoted by Tn that represents the number of dots. Mathematical Formula Below is the mathematical formula for a triangular number: where n is a positive number such as 1, 2, 3, 4, 5, ..... , n. Triangular Number Sequence Below is the triangular number sequence: [1, 3, 6, 10, 15, 21, 28, 36, ..... ] Example For n = 1 Putting this value of n in the above mathematical formula: T1 = 1(1 + 1)/2 T1 = 1(2)/2 T1 = 2/2 T1 = 1 For n = 2 Putting this value of n in the above mathematical formula: T2 = 2(2 + 1)/2 T2 = 2(3)/2 T2 = 6/2 T2 = 3 For n = 3 Putting this value of n in the above mathematical formula: T3 = 3(3 + 1)/2 T3 = 3(4)/2 T3 = 12/2 T3 = 6 For n = 4 Putting this value of n in the above mathematical formu...

One of the amazing concepts associated with the numbers is multiplication with three and adding the resultant digits. As per this amazing concept, if any number is multiplied with three and the resultant digits are added all together, then the new resultant digits are divisible by three. The Concept Step 1: Take any number. Step 2: Multiply the number (taken in Step 1) with three. Step 3: Add the resultant digits (obtained in Step 2) all together. Step 4: The new resultant formed in Step 3 is divisible by three. For Example Number 15 Multiply the number with three: 15 x 3 => 45. Add the resultant digits: 4 + 5 => 9. The new resultant is divisible by three: 9 / 3 => 3. Number 25 Multiply the number with three: 25 x 3 => 75. Add the resultant digits: 7 + 5 => 12. The new resultant is divisible by three: 12 / 3 => 4. Number 99 Multiply the number with three: 99 x 3 => 297. Add the resultant digits: 2 + 9 + 7 => 18. The new resu...

Pronic number, also known as Oblong number, Heteromecic number, or Rectangular number, is a positive integer n which is the product of two consecutive integers. Pronic Number Formula Pronic number formula is given below: n (n + 1) Pronic Number Sequence 0, 2, 6, 12, 20, 30, 42, ….., n. For Example Integer : 56 For integer 56, the value of first consecutive integer = 7. Putting this value of first consecutive integer in Pronic number formula: 7 (7 + 1) => 7 x 8 => 56. Integer : 72 For integer 72, the value of first consecutive integer = 8. Putting this value of first consecutive integer in Pronic number formula: 8 (8 + 1) => 8 x 9 => 72. Integer : 110 For integer 110, the value of first consecutive integer = 10. Putting this value of first consecutive integer in Pronic number formula: 10 (10 + 1) => 10 x 11 => 110. Integer : 156 For integer 156, the value of first consecutive integer = 12. Putting this value of first consecu...

In number theory, a Practical number or Panarithmic number is a positive integer n that can represent all smaller numbers m (m < n) as the sums of distinct divisors of n. In other words, any positive integer n having the property that all smaller integers can be represented as the sums of distinct divisors of integer n is referred to as a Practical number or Panarithmic number. The Procedure Step 1:  Consider a positive integer n. Step 2:  Find out the distinct divisors of n. Step 3:  Check whether all numbers smaller than n can be represented as the sums of distinct divisors of n. Step 4:  If all numbers smaller than n can be represented as the sums of distinct divisors of n, then n is a Practical number or Panarithmic number, else not. Practical Number Sequence 1, 2, 4, 6, 8, 12, 16, 18, 20, ….., n. For Example Integer: 24 Distinct divisors of 24: 1, 2, 3, 4, 6, 8, 12. All smaller numbers such as: 5 => 2 + 3 7 => 3 + 4 9 =>...

The Catalan numbers are a sequence of positive integers that occur in numerous counting problems as far as the combinatorics (the branch of mathematics concerned with counting) is concerned. The Catalan numbers are named after the Belgian mathematician Eugene Charles Catalan [30 May, 1814 – 14 February, 1894]. Below is the formula for finding the Catalan numbers: where n is an integer = 0, 1, 2, 3, 4, 5, 6, 7, ………., n-times. Catalan numbers formula The first few Catalan numbers for different values of integer n are shown below: C 0 = 1. C 1 = 1. C 2 = 2. C 3 = 5. C 4 = 14. C 5 = 42. Hope you have understood this amazing concept of  Catalan numbers. In case of any issue, provide your valuable comments.

This post is about amazing facts which may not be known to you before reading here. So let’s start about these amazing facts: Fact 1: Every odd number contains the letter “e” in it. Fact 2: The only number spelt with letters arranged in alphabetical or ascending order is “forty”. Fact 3: The only number spelt with letters arranged in descending order is “one”. Fact 4: Obelus is the symbol for division. Fact 5: The only number from 0 to 1000 that contains the letter e in it is “one thousand”. Fact 6: 1/100 th part of a second is called jiffy. Fact 7: Temperature -40 o C is equal to -40 o F. [C = Celsius, F = Fahrenheit]. Fact 8: 2 is the only even prime number. Fact 9: Pythagoras’ constant 1.141 is the square root of 2. Fact 10: The only number that cannot be represented in Roma numerals is zero. Fact 11: 6 weeks = 10! seconds. [Proof: 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800 seconds = 60,480 minutes = 1,008 hours = 42 days = 6 weeks]. Fact 12: 111,1...

The concept of Self number or Devlali number or Colombian number was introduced by D. R. Kaprekar (Dattatreya Ramchandra Kaprekar). Self number or Devlali number or Colombian number is an integer that cannot be expressed as the sum of any other integer and its individual digits. In other words, if any integer y cannot be expressed as the sum of any other integer n and its individual digits, then it is referred to as Self number or Devlali number or Colombian number. Important Points Point 1: All integer n less than 15 give the result less than 20. Point 2: All integer n greater than or equal to 15 give the result greater than 20. The Procedure Step 1: Consider an integer y. Step 2: Take any other integer n. Step 3: Add integer n with its individual digits. Step 4: If the sum is not equal to the original integer y taken in Step 1, then integer y is Self number or Devlali number or Colombian number, else not. For Example Integer y: 20 Take any other integer: 1...

D. R. Kaprekar (Dattatreya Ramchandra Kaprekar) introduced an amazing concept Kaprekar number in number theory. Kaprekar number is a natural number whose square when divided into two equal or non-equal parts; left and right, then the sum of both parts must be equal to the original number. None of the parts should have the value zero. The Procedure Step 1: Consider a natural number n. Step 2: Square the number n 2 . Step 3: Divide the value after square into two equal or non-equal parts: left x or x-1 digits and right y digits. Step 4: Add both left and right parts: (x + y) or (x-1 + y). Step 5: If the resultant z is equal to the original number n, then number n is a Kaprekar number, else not. For Example Number n: 9 Square the number n 2 : 9 2 => 81. Divide the value after square into two equal or non-equal parts: 8 and 1. Add both left and right parts: 8 + 1=> 9. Here the resultant 9 is equal to the original number n. Hence, 9 is a Kaprekar number. ...

Lemoine’s conjecture or Levy’s conjecture, named after Emile Lemoine and Hyman Levy, respectively, is a popular conjecture as far as the number theory is concerned. According to this conjecture, every odd positive integer greater than 5 can be expressed as the sum of an odd prime number and an even semiprime. A prime number other than 2 is called an odd prime number . A semiprime is the product of two prime numbers . As per the definition, every odd positive integer n > 5 is the sum of an odd prime p and an even semiprime 2q. Mathematically,  n = sum(p + (2q)) . Conditions Condition 1:  Integer n must be odd. Condition 2:  Integer n must be greater than 5. For Example Integer n: 7 Sum of an odd prime p and even semiprime 2q: 3 + (2 x 2) => 7. Integer n: 11 Sum of an odd prime p and even semiprime 2q: 5 + (2 x 3) => 11. Integer n: 13 Sum of an odd prime p and even semiprime 2q: 7 + (2 x 3) => 13. Integer n: 17 Sum of an odd pri...

Christian Goldbach, a Russian mathematician [March 18, 1690 – November 20, 1764], given a popular conjecture in number theory named as Goldbach’s conjecture . According to this conjecture, every even positive integer greater than 2 can be expressed as the sum of two prime numbers. As per the definition, every even positive integer n > 2 is the sum of two prime numbers p and q. Mathematically, n = sum(p, q) . Conditions Condition 1: Integer n must be even. Condition 2: Integer n must be greater than 2. For Example Integer n: 10 Sum of two prime numbers p and q: 3 + 7 => 10. Integer n: 44 Sum of two prime numbers p and q: 3 + 41 => 44. Integer n: 58 Sum of two prime numbers p and q: 5 + 53 => 58. Integer n: 90 Sum of two prime numbers p and q: 7 + 83 => 90. Integer n: 100 Sum of two prime numbers p and q: 3 + 97 => 100. H ope you have understood this amazing concept of Goldbach’s conjecture.