Sum of digits

Number Theory Level 3

Let f ( n ) f(n) mean the sum of n n 's digits, where n n is a natural number.

For example: f ( 12 ) = 1 + 2 = 3 , f ( 2017 ) = 2 + 0 + 1 + 7 = 10 f(12)=1+2=3, f(2017)=2+0+1+7=10 .

Is the next sentence true or false?

There exist three different positive integers ( x , y , z ) (x, y, z) , such that x + f ( x ) = y + f ( y ) = z + f ( z ) x+f(x)=y+f(y)=z+f(z) .

True False

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

Áron Bán-Szabó
Jun 27, 2017

Yes, there exist. Here is an example:

{ x = 999 999 999 999 891 y = 999 999 999 999 900 z = 1 000 000 000 000 008 \large \begin{cases} x=999\space999\space999\space999\space891 \\ y=999\space999\space999\space999\space900 \\ z=1\space000\space000\space000\space000\space008 \end{cases}

Now x + f ( x ) = y + f ( y ) = z + f ( z ) = 1000000000000017 x+f(x)=y+f(y)=z+f(z)=1 000 000 000 000 017

Using full induction we can prove that for every k k positive integer there exist x 1 , x 2 , x 3 , … , x k x_1, x_2, x_3, \dots, x_k positive natural numbers such that x 1 + f ( x 1 ) = x 2 + f ( x 2 ) = ⋯ = x k + f ( x k ) x_1+f(x_1)=x_2+f(x_2)=\dots=x_k+f(x_k) .

3 Helpful 0 Interesting 0 Brilliant 0 Confused

This is surprising! How did you find these integers?

Pi Han Goh - 3 years, 11 months ago

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