To find the L.C.M. (Least Common Multiple) of the given numbers, we will follow these steps:
### Part (i): Find the L.C.M. of 20 and 24
**Step 1: Factorize the numbers.**
- The prime factorization of 20 is:
\[
20 = 2 \times 2 \times 5 = 2^2 \times 5^1
\]
- The prime factorization of 24 is:
\[
24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1
\]
**Step 2: Identify the highest power of each prime factor.**
- For the prime factor 2, the highest power is \(2^3\) (from 24).
- For the prime factor 3, the highest power is \(3^1\) (from 24).
- For the prime factor 5, the highest power is \(5^1\) (from 20).
**Step 3: Write the L.C.M. using the highest powers.**
\[
\text{L.C.M.} = 2^3 \times 3^1 \times 5^1
\]
**Step 4: Calculate the L.C.M.**
\[
2^3 = 8, \quad 3^1 = 3, \quad 5^1 = 5
\]
Now, multiply these together:
\[
8 \times 3 = 24
\]
\[
24 \times 5 = 120
\]
Thus, the L.C.M. of 20 and 24 is **120**.
### Part (ii): Find the L.C.M. of 16, 22, and 40
**Step 1: Factorize the numbers.**
- The prime factorization of 16 is:
\[
16 = 2 \times 2 \times 2 \times 2 = 2^4
\]
- The prime factorization of 22 is:
\[
22 = 2 \times 11 = 2^1 \times 11^1
\]
- The prime factorization of 40 is:
\[
40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5^1
\]
**Step 2: Identify the highest power of each prime factor.**
- For the prime factor 2, the highest power is \(2^4\) (from 16).
- For the prime factor 11, the highest power is \(11^1\) (from 22).
- For the prime factor 5, the highest power is \(5^1\) (from 40).
**Step 3: Write the L.C.M. using the highest powers.**
\[
\text{L.C.M.} = 2^4 \times 11^1 \times 5^1
\]
**Step 4: Calculate the L.C.M.**
\[
2^4 = 16, \quad 11^1 = 11, \quad 5^1 = 5
\]
Now, multiply these together:
\[
16 \times 11 = 176
\]
\[
176 \times 5 = 880
\]
Thus, the L.C.M. of 16, 22, and 40 is **880**.
### Summary of Results:
- L.C.M. of 20 and 24 is **120**.
- L.C.M. of 16, 22, and 40 is **880**.
