How many things do you know about digits sum?

Number Theory Level pending

There exists a natural number n n such that s ( n ) = 10 s(n) = 10 and s ( n 2 ) = 100 s(n^2) = 100 , where s ( n ) s(n) is the sum of digits of n n ?

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Anca Baltariga by Anca Baltariga , 26, Romania

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1 solution

Anca Baltariga
Dec 20, 2016

Let n = 1 0 2 0 + 1 0 2 1 + . . . + 1 0 2 9 n=10^{2^0} + 10^{2^1}+...+10^{2^9} . Then s ( n ) = 10 s(n)=10 and n 2 = 1 0 2 1 + 1 0 2 2 + . . . + 1 0 2 10 + 2 ⋅ 1 0 2 0 + 2 1 + 2 ⋅ 1 0 2 0 + 2 2 + . . . + 2 ⋅ 1 0 2 8 + 2 9 n^2 = 10^{2^1}+10^{2^2}+...+10^{2^{10}} + 2\cdot 10^{2^0+2^1} +2\cdot10^{2^0+2^2}+...+2\cdot 10^{2^8+2^9} . It is easy to see that s ( n 2 ) = 10 ⋅ 1 + 10 ⋅ 9 2 ⋅ 2 = 100 s(n^2)=10\cdot 1+\frac{10\cdot 9}{2}\cdot 2=100 .

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An interesting further question would be: what would be the smallest possible n? I think I have the answer...

Paul Hindess - 4 years, 5 months ago

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Post it as another problem !

Pi Han Goh - 4 years, 5 months ago

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