The Venn diagram shows the prime factors of 12 (F12) and y (Fy).

The Venn diagram shows the prime factors of 12 (F12) and y (Fy).
The prime factors of 12 are $2 \times 2 \times 3$.
From the diagram, the F12 circle contains $x$, and the common factors (intersection) are $2, 3$.
So, the prime factors of 12 are $x, 2, 3$. For this to be $2, 2, 3$, the value of $x$ must be 2.
The prime factors of y (Fy circle) include the common factors $2, 3$ and additional factors $2, 3$ in the Fy-only section.
So, the prime factors of y are $2, 3, 2, 3$.
Therefore, $y = 2 \times 3 \times 2 \times 3 = 36$.

• a) Find the value of:
  • (i) x: $$x = \boxed{2}$$
  • (ii) y: $$y = \boxed{36}$$
• b) Find the GCF of 12 and y. The prime factors of 12 are $2^2 \times 3^1$. The prime factors of $y=36$ are $2^2 \times 3^2$. The GCF is the product of the lowest powers of common prime factors. $$\text{GCF}(12, 36) = 2^2 \times 3^1 = 4 \times 3 = \boxed{12}$$
• c) Find the LCM of 12 and Y. The LCM is the product of the highest powers of all prime factors. $$\text{LCM}(12, 36) = 2^2 \times 3^2 = 4 \times 9 = \boxed{36}$$

Find the square of the following numbers.

• a) 9: $$9^2 = 9 \times 9 = \boxed{81}$$
• b) 25: $$25^2 = 25 \times 25 = \boxed{625}$$
• c) 16: $$16^2 = 16 \times 16 = \boxed{256}$$

Find the square root of the following numbers.

• a) 144: $$\sqrt{144} = \boxed{12}$$
• b) 196: $$\sqrt{196} = \boxed{14}$$
• c) 441: $$\sqrt{441} = \boxed{21}$$
• d) 400: $$\sqrt{400} = \boxed{20}$$