Solve (1)

Algebra Level 2

Denote by S ( x ) S(x) the sum of the digits of a positive integer x x .

Solve for x x in the given expression: x + S ( x ) + S ( S ( x ) ) = 1993 x +S(x)+S(S(x))=1993

no solution 1990 1993 1992 1991

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1 solution

David Vreken
Oct 7, 2018

By casting out nines , the remainder of a number divided by 9 9 is the same as the remainder of a number's digital sum divided by 9 9 . In other words, x S ( x ) S ( S ( x ) ) ( m o d 9 ) x \equiv S(x) \equiv S(S(x)) \pmod{9} , which means that x + S ( x ) + S ( S ( x ) ) ( m o d 9 ) x + S(x) + S(S(x)) \pmod{9} must be divisible by 3 3 .

However, 1993 ( m o d 9 ) = 4 1993 \pmod{9} = 4 , but 4 4 is not divisible by 3 3 , so there is no solution to x + S ( x ) + S ( S ( x ) ) = 1993 x + S(x) + S(S(x)) = 1993 .

Thank you, nice solution.

Hana Wehbi - 2 years, 8 months ago

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