The Unique Natural Number: 18 Twice the Sum of Its Digits

Among the infinite set of natural numbers, there is one number that stands out due to its unique property. This number, 18, is the only natural number that equals twice the sum of its digits. Let's explore why this is the case and delve into the fascinating realm of number theory.

Understanding the Problem

We are looking for a natural number ( n ) such that:

( n 2 times S ), where ( S ) is the sum of the digits of ( n ).

Step 1: Single-Digit Numbers

Let's begin with single-digit numbers. Any single-digit number ( n ) is its own sum of digits, so:

( n S )

Substituting into the equation:

( n 2 times n )

This simplifies to:

( n 2n )

The only solution to this equation is ( n 0 ), but 0 is not considered a natural number. Therefore, it's clear that no single-digit numbers satisfy the condition.

Step 2: Two-Digit Numbers

Now, let's look at two-digit numbers. A two-digit number ( n ) can be represented as ( 10a b ), where ( a ) is the tens digit and ( b ) is the units digit. Hence, the sum of the digits ( S ) is:

( S a b )

The equation becomes:

( 10a b 2 times (a b) )

Simplifying this, we get:

( 10a b 2a 2b )

This can be rearranged to:

( 8a b )

Since ( b ) must be a single digit (0 to 9) and ( a ) must be a non-zero digit (1 to 9), the only possible value for ( a ) is 1. Substituting ( a 1 ), we find:

( b 8 )

Hence, the only number that satisfies the condition for two-digit numbers is:

( n 18 )

Step 3: Larger Numbers

Let's now consider three-digit numbers and numbers with more digits. A three-digit number can be written as ( 100a 10b c ), where ( a, b, c ) are its digits. The sum of its digits ( S ) is:

( S a b c )

The equation becomes:

( 100a 10b c 2 times (a b c) )

Rearranging terms, we get:

( 100a 10b c 2a 2b 2c )

This simplifies to:

( 98a 8b c )

For any valid digits of ( a ) and ( b ) (1 to 9 and 0 to 9, respectively), ( c ) becomes impossibly large, exceeding the possible range of 0 to 9. Similarly, for numbers with more digits, the left side grows much faster than twice the sum of the digits, making it impossible to satisfy the equation.

Conclusion

The only natural number that satisfies the condition is 18. This result is unique and fascinating, as no other number in the natural numbers has the same property. The exploration of such properties enhances our understanding of number theory and showcases the intricate relationships between numbers and their digits.