Problem Solving

Let f(x) denotes the sum of the digits of the positive integer x. For example, f(8) =8, f(123) = 1+2+3=6. For how many two-digit values of x is f(f(x))=3?

A) 3
B) 4
C) 6
D) 9
E) 10

The OA is E.

Can any expert help me with this PS question please? I don't have it clear. Thanks.

Hello LUANDATO.

It is a really interesting question.

The idea is that we must find the numbers of two digits such that the sum of their digits is:

- 3, that is to say, f(x)=3, so that f(f(x))=f(3)=3.
- 12, that is to say, f(x)=12, so that f(f(x))=f(12)=3.

The numbers that satisfy this property are: 12, 21, 30, 39, 48, 57, 66, 75, 84 and 93.

f(f(12))=f(3)=3
f(f(21))=f(3)=3
f(f(30))=f(3)=3
f(f(39))=f(12)=3
f(f(48))=f(12)=3
f(f(57))=f(12)=3
f(f(66))=f(12)=3
f(f(75))=f(12)=3
f(f(84))=f(12)=3
f(f(93))=f(12)=3

So, the answer is E.

Note that f(f(3))=3, but 3 is one-digit number, that is why it is not on the list above.

I hope this can help you to understand how to solve this kind of questions.