Can you solve this equation involving the sum of digits of a positive integer?

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SUMMARY

The equation to solve is x + P(x) + P(P(x)) = 1993, where P(x) denotes the sum of the digits of the positive integer x. The discussion emphasizes the need to evaluate the properties of digit sums and their implications on the integer solutions. The solution involves iterating through potential values of x and calculating P(x) and P(P(x)) until the equation holds true.

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If $P(x)$ represents the sum of the digits of a positive integer $x$. Solve $x+P(x)+P(P(x))=1993$.
 
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anemone said:
If $P(x)$ represents the sum of the digits of a positive integer $x$. Solve $x+P(x)+P(P(x))=1993$.

x mod 9 = P(x) mod 9 = P(P(x)) mod 9 = say k

so x + P(x) + P(P(x)) = 3 k mod 9

but x + P(x) + P(P(x)) mod 9 = 4 mod 9

as 3 is a factor of 9 there is no y such that 3y = 4 mod 9

so no solution
 

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