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RMO 2014

For any positive integer \(n\) , let \(S(n)\) denote the sum of digits of \(n\). Find the number of \(3\) digit natural numbers \(n\) for which \(S(S(n))=2\). What is the answer ?

Note by Arushi Goel
2 years, 9 months ago

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100 Anshul Sanghi · 2 years, 9 months ago

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The crux move is to notice all such numbers are congruent \(2\) modulo \(9\) Souryajit Roy · 2 years, 9 months ago

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Answer is \(\boxed{100}\) Karthik Sharma · 2 years, 9 months ago

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@Karthik Sharma Can i know how,it came 100? i have tried the question but couldn't get the answer Nishant Singh · 1 year, 10 months ago

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@Nishant Singh First start with taking a number \(abc\) where a,b and c are digits.

and let \(a+b+c = xy\) Where \(xy\) is a number and x and y are digits.

So, we have \(S(S(n)) = S(xy) = x+y = 2\)

Then make three cases - x=0, y=2 , x=2, y=0 and x=y=1.

Do a little bit of P & C to get the answer. Good luck.

(I did this question in RMO 2014 the same way I mentioned. There are other quicker methods to do this too.) Karthik Sharma · 1 year, 8 months ago

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