# Necessary and sufficient condition on whether there exists a multiple satisfying sum of integers property

Let $$a$$ and $$b$$ be positive integers,
give a necessary and sufficient condition on $$b$$ so that there exists an integer $$k$$ such that $$S(ka)=b$$
where $$S(x):=$$ the sum of digits of $$x$$ in base $$10$$

• What have you tried and where did you find this problem? If you list your attempts in the question we will be better able to help you. – Carl Schildkraut Oct 22 '18 at 20:37
• I've tried using congruence modulo $9$ because $S(x)=x[9]$ for all $x$ but it wasn't enouph – K.Elias Oct 22 '18 at 21:40