In a set we have $a(b+c)=ab+c$. What is it?

Question

suppose $A\subseteq \mathbb{N}$ and for any $a,b,c\in A$ with $a<b<c$ we have $$a(b+c)=ab+c$$ what are all $A$ with this property?!

here $\mathbb{N}=\{1,2,3,...\}$.

Answer 1

Apply the distributive property, subtract the common term, and you get $ac=c$, so $a=1$. Any three element subset that includes $1$ will force this. Any set with less than three elements will also meet the requirement as there is no $a,b,c$.