Question

What are Armstrong numbers?

Answer 1

Hint: First of all, we must know the definition of Armstrong number. So, in order to know about it let N be any number. Determine how many digits are in the number. Let us assume that it is n. now take each digit in the number and raise it to the n power. After calculating the $n^{th}$ power of each digit add all of them. If we get the sum as the original number then the original number is called Armstrong number.

Complete step by step solution:
An Armstrong number is defined as the sum of $n^{th}$ power of each digit to a n digit number is equal to that number. Let $N=abc$ is a three number which is an Armstrong number, so the mathematical form of the number is $N={{a}^{3}}+{{b}^{3}}+{{c}^{3}}$, if $N=abcd$ be a four digit number then we have $N={{a}^{4}}+{{b}^{4}}+{{c}^{4}}+{{d}^{4}}$, similarly for five, six, seven and so on…
Let us explain the definition using the example.
Suppose we take a number $N=153$ which is three-digit number,
Now we calculate
\[\begin{align}
  & {{1}^{3}}+{{5}^{3}}+{{3}^{3}} \\
 & =1+125+27 \\
 & =153 \\
\end{align}\]
So, we can write
$153={{1}^{3}}+{{5}^{3}}+{{3}^{3}}$
Hence, we can say that 153 is an Armstrong number.
Take another number of four digit $N=1634$
Now here we calculate
\[\begin{align}
  & {{1}^{4}}+{{6}^{4}}+{{3}^{4}}+{{4}^{4}} \\
 & =1+1296+81+256 \\
 & =1634 \\
\end{align}\]
So, we can write
\[1634={{1}^{4}}+{{6}^{4}}+{{3}^{4}}+{{4}^{4}}\]
Hence, we can say that the number 1634 is also an Armstrong number.
So, the above illustration explains the definition of Armstrong number.
Note: it should be noted that this number is also known as a Narcissistic number as well as plus perfect number. As per the definition all one-digit numbers are Armstrong numbers. There are no any two digits Armstrong numbers. Three-digit Armstrong numbers are 153,370,371 and 407.