3. Let $a$ and $b$ be natural numbers such that $\text{LCM}(a, b)=10$ and $\text{GCD}(a, b+2)=12$. Calculate

$$
\text{GCD}(a, 2b) + \text{GCD}(a, 3b).
$$
3. From $\gcd(a, b)=10$ we conclude that 5 is the greatest odd common divisor of $a$ and $b$, so 5 is also the greatest odd common divisor of $a$ and $2b$. Therefore, to calculate $\gcd(a, 2b)$, we need to determine the highest power of 2 that divides both $a$ and $2b$. From $\gcd(a, b)=10$ we conclude that the numbers $a$ and $b$ are divisible by 2, but not both by 4. On the other hand, from $\gcd(a, b+2)=12$ we conclude that $a$ is divisible by 4, so $b$ is not divisible by 4. Therefore, $a$ and $2b$ are divisible by 4, but $2b$ is not divisible by 8, so $\gcd(a, 2b)=4 \cdot 5=20$.

Similarly, from $\gcd(a, b)=10$ it follows that $a$ and $b$ are not simultaneously divisible by 3, and from $\gcd(a, b+2)=12$ it follows that $a$ is divisible by 3, so $b$ is not divisible by 3. Therefore, $a$ and $3b$ are divisible by 3, but $3b$ is not divisible by 9, so from this and $\gcd(a, b)=10$ we get $\gcd(a, 3b)=30$.

Therefore, $\gcd(a, 2b)+\gcd(a, 3b)=20+30=50$.