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Problem B. 4072. (March 2008)

B. 4072. Let S(n) denote the sum of the digits of the natural number n. Prove that there are infinitely many natural numbers n not ending in 0, such that S(n2)=S(n).

Suggested by G. Holló, Budapest

(3 pont)

Deadline expired on April 15, 2008.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás: Minden k természetes számra n=10k-1 megfelelő lesz. Ekkor ugyanis egyrészt n olyan k-jegyű szám, amelynek minden számjegye 9-es, vagyis S(n)=9k. Másrészt

n2=102k-2.10k+1=10k(10k-2)+1,

vagyis S(n2)=S(10k-2)+1. Itt 10k-2 az a k-jegyű szám, amelynek minden számjegye 9-es, kivéve az utolsót, ami 8-as. Ezért S(10k-2)=9(k-1)+8, S(n2)=9(k-1)+8+1=9k=S(n).


Statistics:

118 students sent a solution.
3 points:112 students.
2 points:4 students.
1 point:1 student.
Unfair, not evaluated:1 solution.

Problems in Mathematics of KöMaL, March 2008