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. `2^3 , 3200`

Answer

To find the H.C.F. (Highest Common Factor) and L.C.M. (Least Common Multiple) of the numbers \(2^4\), \(8^2\), \(16^2\), and \(20^3\), we will follow these steps:

### Step 1: Express all numbers in terms of their prime factors.
- \(2^4\) is already in prime factor form.
- \(8^2 = (2^3)^2 = 2^{3 \times 2} = 2^6\)
- \(16^2 = (2^4)^2 = 2^{4 \times 2} = 2^8\)
- \(20^3 = (2^2 \times 5)^3 = 2^{2 \times 3} \times 5^3 = 2^6 \times 5^3\)

So, we have:
- \(2^4\)
- \(2^6\)
- \(2^8\)
- \(2^6 \times 5^3\)

### Step 2: Find the H.C.F.
The H.C.F. is found by taking the lowest power of each prime factor present in all numbers.

- For \(2\): The powers are \(4\), \(6\), \(8\), and \(6\). The lowest power is \(2^4\).
- For \(5\): It appears only in \(20^3\) as \(5^3\), so it does not contribute to the H.C.F.

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

### Step 3: Find the L.C.M.
The L.C.M. is found by taking the highest power of each prime factor present in any of the numbers.

- For \(2\): The powers are \(4\), \(6\), \(8\), and \(6\). The highest power is \(2^8\).
- For \(5\): The highest power is \(5^3\) (from \(20^3\)).

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

### Step 4: Calculate the values.
- \(2^8 = 256\)
- \(5^3 = 125\)

Now, calculate the L.C.M.:
\[ \text{L.C.M.} = 256 \times 125 \]

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

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