If A =factors of 24} and B={factors of 36}, find, (i) `A cap B`, (ii) `A cup B`

Answer

To solve the problem, we need to find the intersection and union of the sets A and B, where:

- A = {factors of 24}
- B = {factors of 36}

### Step 1: Find the factors of 24
The factors of 24 are the numbers that divide 24 without leaving a remainder. The factors of 24 are:
- 1 (24 ÷ 1 = 24)
- 2 (24 ÷ 2 = 12)
- 3 (24 ÷ 3 = 8)
- 4 (24 ÷ 4 = 6)
- 6 (24 ÷ 6 = 4)
- 8 (24 ÷ 8 = 3)
- 12 (24 ÷ 12 = 2)
- 24 (24 ÷ 24 = 1)

Thus, the set A is:
\[ A = \{1, 2, 3, 4, 6, 8, 12, 24\} \]

### Step 2: Find the factors of 36
Similarly, we find the factors of 36:
- 1 (36 ÷ 1 = 36)
- 2 (36 ÷ 2 = 18)
- 3 (36 ÷ 3 = 12)
- 4 (36 ÷ 4 = 9)
- 6 (36 ÷ 6 = 6)
- 9 (36 ÷ 9 = 4)
- 12 (36 ÷ 12 = 3)
- 18 (36 ÷ 18 = 2)
- 36 (36 ÷ 36 = 1)

Thus, the set B is:
\[ B = \{1, 2, 3, 4, 6, 9, 12, 18, 36\} \]

### Step 3: Find A intersection B (A ∩ B)
The intersection of sets A and B includes all elements that are present in both sets.

Comparing the two sets:
- A: {1, 2, 3, 4, 6, 8, 12, 24}
- B: {1, 2, 3, 4, 6, 9, 12, 18, 36}

The common elements are:
- 1
- 2
- 3
- 4
- 6
- 12

Thus, the intersection A ∩ B is:
\[ A \cap B = \{1, 2, 3, 4, 6, 12\} \]

### Step 4: Find A union B (A ∪ B)
The union of sets A and B includes all elements that are in either set A or set B or in both.

Combining the elements from both sets without repetition:
- From A: {1, 2, 3, 4, 6, 8, 12, 24}
- From B: {1, 2, 3, 4, 6, 9, 12, 18, 36}

The combined set is:
\[ A \cup B = \{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36\} \]

### Final Answers:
(i) \( A \cap B = \{1, 2, 3, 4, 6, 12\} \)

(ii) \( A \cup B = \{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36\} \)

---