Hi! Hope you’re doing fine.
I just wanted to ask some question regarding Manhattan exercices.
I got confused with these two tips of questions:

The number of distinct positive factors of 10 vs the number of distinct prime factors of 210.
I split both numbers in their primes, 10 is 5x2 (add 1 to each exponent) so the result is 4 and 210 is 3x7x5x2 and did the same trick, but the answer was 4. I´m a bit confused were to use or not this trick.

LCM of 22 and 6 vs GCF of 66 and 99.
I also reduced the numbers to their primes factors so 22 is 11x2 and 6 is 3x2, so we said that the LCM is the biggest number of each prime, so in this case was 2 and I thought the other should work the same. I don´t understand why in these cases the trick doesn´t work.
Regarding your first question, I think it’s just a problem with your foundations (concepts). Recall that not all numbers have the same number of positive factors as the number of prime factors. Notice the emphasis that I am doing. So, take for example the number 8. How many positive factors does 8 have? You applied the correct trick, so you prime factorize 8 and get that 8=2^3, and you just add one to the exponent. Thus, 8 has 4 different positive factors. What about the number of distinct prime factors of 8? You basically prime factorize as well, so we already know that 8=2^3. Notice that 2 is the only primer factor, so 8 has only one distinct prime factor.
So when you are doing 3\cdot 7 \cdot 5 \cdot 2 and “do the same trick”, you are actually finding the number of distinct positive factors, not the number of different prime factors, because in this case, the prime factors would only be 3, 7, 5, 2, so there is a total of 4 distinct prime factors.
Regarding your second question, recall that the LCM is the least common multiple, so in this case, indeed you need to prime factorize both numbers, and take the greatest exponent of each factor between the two numbers. So, 22=2\cdot 11 and 6=3\cdot 2, so just compare the exponents of each prime factors. How many 2’s are in 22? One. How many 2’s are in 6? One. Then, keep that 2. How many 3’s are in 22? Zero. How many 3’s are in 6? One. Then, keep that 3. How many 11’s are in 22? One. How many 11’s are in 6? Zero. Then keep that one 11. After doing this, just multiply the numbers, so 2\cdot 3 \cdot 11=66 and 66 is the LCM between the two numbers.
The GCF might sound similar to the LCM, but it is not the same concept. The GCF is the Greatest Common Factor i.e. the greatest factor that both numbers share. Again, prime factorize 66 and 99, so we know that 66=2 \cdot 3 \cdot 11 and 99= 3^2 \cdot 11. Now compare each factor. How many 2’s does 66 have? One. What about 99? Zero. Thus, there is a 2 in 66 but not in 99, so 2 cannot be a common factor. Notice that you could call it a shared factor if that makes the name more familiar. What about 3’s? There is one 3 in 66 and two 3’s in 99, so both numbers share one 3. Keep that 3. What about 11? Both numbers share an 11, so keep that number as well. After you finish comparing all the prime factors, multiply the ones you set apart, in this case, 3\cdot 11= 33 so 33 is our GCF.