# Number of solutions for $a+b+c+ab+bc+ac+abc=29$ [closed]

Let the number N be the smallest three digit number with digits $$a,b,c$$. $$a+b+c+ab+bc+ca +abc=29$$. Evaluate $$\frac{N+1}{5}$$.

By adding $$1$$ on both sides we get $$(a+1)(b+1)(c+1)=30$$. How should I proceed further and is there any way without hit and trial?

We want to find the smallest $$N$$ possible, so minimise each digit one at a time.
$$a \ne 0$$ so the smallest $$a$$ is $$1$$. This gives $$(b+1)(c+1) = 15$$ and you can continue from here.