Let's break down the tasks from the image you've uploaded.

(b) Write down the six prime numbers in the universal set $\mathcal{U}$.

The universal set in the diagram appears to have the elements: $2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16$.

Out of these, the prime numbers are the ones that have only two distinct positive divisors: 1 and the number itself.

The six prime numbers in the universal set are:
2, 3, 5, 7, 11, 13.

(c) Complete the Venn diagram to show the elements of these sets.

• Set $M = \{ x : x \text{ is a multiple of 3} \}$ will contain numbers that are divisible by 3: $3, 6, 9, 12, 15$.
• Set $F = \{ x : x \text{ is a factor of 15} \}$ will contain the factors of 15: $1, 3, 5, 15$.

Let's break down where each number goes:

• The number 9 is both a multiple of 3 (so it belongs to $M$) and a factor of 15 (so it belongs to $F$).
• The number 1 is a factor of 15 (belongs to $F$).
• The number 3 is both a multiple of 3 (belongs to $M$) and a factor of 15 (belongs to $F$).
• The number 5 is a factor of 15 (belongs to $F$).
• The number 6 is a multiple of 3 (belongs to $M$).
• The number 7 does not fit in either set, so it remains outside.
• The number 8 is neither a multiple of 3 nor a factor of 15, so it stays outside as well.
• The number 11 is neither a multiple of 3 nor a factor of 15, so it stays outside as well.
• The number 12 is a multiple of 3 (belongs to $M$).
• The number 13 does not fit in either set and stays outside.
• The number 14 is neither a multiple of 3 nor a factor of 15, so it stays outside.
• The number 15 is both a multiple of 3 and a factor of 15, so it goes in both $M$ and $F$.

Updated Venn diagram:

• M only: $3, 6, 12$
• F only: $1, 5$
• M ∩ F (both M and F): $9, 15$
• Outside both M and F: $2, 7, 8, 11, 13, 14$

(i) Complete the Venn diagram:
You can now place the numbers according to the breakdown above.

Would you like me to move on to the next parts of the problem? Feel free to ask for further clarification! Here are some related questions you may find interesting:

1. What are the properties of prime numbers?
2. How do you find the factors of a number?
3. How do you find the multiples of a number?
4. Can the universal set contain both prime numbers and factors of a number?
5. What are the different set operations like union, intersection, and difference?

Tip: Remember that factors of a number are all the whole numbers that divide it exactly (without leaving a remainder). Multiples of a number are all numbers that can be obtained by multiplying it by an integer.