Armstrong number or narcissistic number

Q. What is Armstrong number or

     narcissistic number?

Ans: A number x is an Armstrong number or narcissistic number if it equal to the sum of its own digits raised to the power of the number of digits.
               

or,

An Armstrong number is any number of n-digits which is equal to the sum of nth power of digits in the number. 

For example -  If we take 371 , it is an Armstrong number as the number of digits (=n) here is n=3, so

371 = 3³ + 7³ + 1³ = 27 + 343 + 1 = 371

Ex- If we take 1634 ,
number of digits in 1634 is 4, so

1634=1⁴+6⁴+3⁴+4⁴=1+1296+81+256=1634
hence , 1634 is an Armstrong number.


Ex- If we take 54748 ,
number of digits in 54748 is 5, so

54748=5⁵+4⁵+7⁵+4⁵+8⁵=3125+1024+16807+1024+32768=54748
hence, 54748 is an Armstrong number.

Some important points :

• All single digit numbers are       Armstrong numbers.

For example :  let us take a number 4, here number of digits in 4 is 1, so

  4 = 4¹ = 4 ⇒Armstrong number

Similarly  0,1,2,3,4,5,6,7,8,9 all are Armstrong numbers.

• There are no 2-digits Armstrong numbers.

• There are four 3-digits Armstrong numbers.

 ex- 153 , 370 , 371 , 407

• There are three 4-digits Armstrong numbers.

ex- 1634 , 8208 , 9474

• There are three 5-digits Armstrong numbers.

ex- 54748 , 92727 , 93084

• There are one 6-digits Armstrong numbers.

ex- 548834

• There are four 7-digits Armstrong numbers.

ex- 1741725 , 4210818 , 9926315 , 9800817

• There are three 8-digits Armstrong numbers.

ex- 24678050 , 24678051 , 88593477

• There are only 88 Armstrong numbers .

• The largest Armstrong number is a 39-digit number which is given below

115132219018763992565095597973971522401



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