Let's solve the tasks step by step.

Prime factorization and HCF:

For 36:

Prime factorization of $36$:

• $36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$.

For 66:

Prime factorization of $66$:

• $66 = 2 \times 3 \times 11$.

Common factors:

• $2$ and $3$.

HCF:

• $HCF = 2 \times 3 = 6$.

For 490:

Prime factorization of $490$:

• $490 = 2 \times 5 \times 7 \times 7 = 2 \times 5 \times 7^2$.

For 420:

Prime factorization of $420$:

• $420 = 2 \times 2 \times 3 \times 5 \times 7 = 2^2 \times 3 \times 5 \times 7$.

Common factors:

• $2$, $5$, and $7$.

HCF:

• $HCF = 2 \times 5 \times 7 = 70$.

Quiz Night Problem:

Greatest number of people in each team (HCF of 30 and 36):

Prime factorization of $30$:

• $30 = 2 \times 3 \times 5$.

Prime factorization of $36$:

• $36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$.

Common factors:

• $2$ and $3$.

HCF:

• $HCF = 2 \times 3 = 6$.

The greatest number of people Joe can put in each team is 6 people.

Would you like further explanation of any part?

Here are 5 related questions:

1. What is the definition of prime factorization?
2. How is the HCF used in real-world scenarios?
3. What is the difference between HCF and LCM?
4. Can two numbers have an HCF of 1? Why or why not?
5. What happens if one of the numbers is a prime number?

Tip: Always write prime factorization systematically, starting with the smallest prime.