# 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
3 years, 7 months ago

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100

- 3 years, 7 months ago

The crux move is to notice all such numbers are congruent $$2$$ modulo $$9$$

- 3 years, 7 months ago

Answer is $$\boxed{100}$$

- 3 years, 7 months ago

Can i know how,it came 100? i have tried the question but couldn't get the answer

- 2 years, 8 months ago

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.)

- 2 years, 6 months ago