The H.C.F. and L.C.M. of `2^4, 8^2, 16^2, 20^3` are :

Options

A. `2^3, 32000`
B. `2^4, 32000`
C. `2^4, 25600`
D. `2^2, 3200`

Answer

To find the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) of the numbers \(2^4\), \(8^2\), \(16^2\), and \(20^3\), we will first express each number in its prime factorization form.

### Step 1: Prime Factorization of Each Number

1. **For \(2^4\)**:
\[ 2^4 = 2 \times 2 \times 2 \times 2 \]

2. **For \(8^2\)**:
\[ 8 = 2^3 \quad \text{thus,} \quad 8^2 = (2^3)^2 = 2^{3 \times 2} = 2^6 \]

3. **For \(16^2\)**:
\[ 16 = 2^4 \quad \text{thus,} \quad 16^2 = (2^4)^2 = 2^{4 \times 2} = 2^8 \]

4. **For \(20^3\)**:
\[ 20 = 2^2 \times 5^1 \quad \text{thus,} \quad 20^3 = (2^2 \times 5^1)^3 = 2^{2 \times 3} \times 5^{1 \times 3} = 2^6 \times 5^3 \]

### Step 2: List the Prime Factorizations
Now we have:
- \(2^4\)
- \(2^6\)
- \(2^8\)
- \(2^6 \times 5^3\)

### Step 3: Finding H.C.F.
To find the H.C.F., we take the lowest power of each prime factor present in all numbers.

- For \(2\): The lowest power is \(2^4\).
- For \(5\): It is not present in \(2^4\), \(2^6\), or \(2^8\), so we do not consider it.

Thus, the H.C.F. is:
\[ \text{H.C.F.} = 2^4 = 16 \]

### Step 4: Finding L.C.M.
To find the L.C.M., we take the highest power of each prime factor present in any of the numbers.

- For \(2\): The highest power is \(2^8\).
- For \(5\): The highest power is \(5^3\).

Thus, the L.C.M. is:
\[ \text{L.C.M.} = 2^8 \times 5^3 \]

Calculating \(2^8\) and \(5^3\):
- \(2^8 = 256\)
- \(5^3 = 125\)

Now, multiply these together:
\[ \text{L.C.M.} = 256 \times 125 \]

Calculating \(256 \times 125\):
\[ 256 \times 125 = 32000 \]

### Final Answers
- H.C.F. = 16
- L.C.M. = 32000