A factory produced an original calculator that performs two operations:

- the usual addition +
- the operation $\circledast$

We know that for any natural number $a$ we have:

$$
\text { (i) } a \circledast a=a \quad \text { and (ii) } a \circledast 0=2 a
$$

and, for any four naturals $a, b, c$ and $d$

$$
\text { (iii) }(a \circledast b)+(c \circledast d)=(a+c) \circledast(b+d) \text {. }
$$

What are the results of the operations $(2+3) \circledast(0+3)$ and $1024 \circledast 48$?
To calculate $(2+3) \circledast(0+3)$, we will use property (iii), and we have:

$$
(2+3) \circledast(0+3)=(2 \circledast 0)+(3 \circledast 3)
$$

Now, by (i) we have $2 \circledast 0=2 \times 2=4$, and by (ii) we have $3 \circledast 3=3$. Therefore,

$$
(2+3) \circledast(0+3)=4+3=7
$$

Now, to calculate $1024 \circledast 48$ we will use the same strategy as above. For this, note that $1024=976+48$ and $48=0+48$.

$$
\begin{aligned}
1024 \circledast 48 & =(976+48) \circledast(0+48) \\
& =(976 \circledast 0)+(48 \circledast 48) \\
& =2 \times 976+48 \\
& =1952+48=2000
\end{aligned}
$$