Denote by the sum of the digits of a positive integer .
Solve for in the given expression:
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
By casting out nines , the remainder of a number divided by 9 is the same as the remainder of a number's digital sum divided by 9 . In other words, x ≡ S ( x ) ≡ S ( S ( x ) ) ( m o d 9 ) , which means that x + S ( x ) + S ( S ( x ) ) ( m o d 9 ) must be divisible by 3 .
However, 1 9 9 3 ( m o d 9 ) = 4 , but 4 is not divisible by 3 , so there is no solution to x + S ( x ) + S ( S ( x ) ) = 1 9 9 3 .