Knowing that the sum $1014_p + 216_p = 1232_p$, for a given base p, is correct. Which base is p?

Question

I would like a lot to know what exactly I have to study to solve this kind of question. That was from my discrete math class, but I've been missing classes, so I don't know which subject is it exactly.

Again, knowing what I can google so I can study to solve that question is much more important than the answer itself. I'd be glad if you could help me.

Answer 1

I'd learn about how to convert different bases to solve this.

To find out what p is, we need an equation that somehow involves it. Remember that to convert a number from any base to base 10, you add each of its digits multiplied by $base^{place - 1}$, for example $$123_8 = 1 \cdot 8^2 + 2 \cdot 8^1 + 3 \cdot 8^0 = 64 + 16 + 3 = 83$$ in base 10. This is perfect since we can form an equation that involves the base with this.

$$1014_p + 216_p = 1232_p$$ $$1 \cdot p^3 + 0 \cdot p^2 + 1 \cdot p^1 + 4 \cdot p^0 + 2 \cdot p^2 + 1 \cdot p^1 + 6 \cdot p^0 = 1 \cdot p^3 + 2 \cdot p^2 + 3 \cdot p^1 + 2\cdot p^0$$ $$2p + 10 = 3p + 2$$ $$\fbox{p = 8}$$

Answer 2

You have $(4+p+p^3)+(6+p+2p^2)=2+3p+2p^2+p^3$ from which $2p+10=3p+2$ then $p=8$ and the represented number is $666$.