Subject 3 Consider the numbers $a=8^{222}-3 \cdot 4^{332}-2^{663} \quad$ and $b=3^{500}-2 \cdot 3^{499}-2 \cdot 3^{498}-\ldots-2 \cdot 3^{443}-2 \cdot 3^{442}$

a) Compare the numbers $a$ and $b$.

b) Determine the smallest two-digit prime number $p$ such that $a+b+p$ is divisible by 10.
## Solution and Scoring

a) $a=2^{633} \quad 1 \text{p}$ $b=3^{500}-(3-1) \cdot 3^{499}-(3-1) \cdot 3^{498}-\cdots-(3-1) \cdot 3^{442} \quad 1 \text{p}$

$b=3^{442} \quad 1 \text{p}$

$a<b \quad 1 \text{p}$

b) $\operatorname{UC}\left(2^{663}\right)=8 ; \quad U C\left(3^{442}=9 \quad 1 \text{p}\right.$

$U C(a+b+p)=0 \Rightarrow U(p)=3 \quad 1 \text{p}$

$p=13 \quad 1 \text{p}$