Find the H.C.E. and L.C.M. of `6!,7!,8!`.

Answer

To find the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) of \(6!\), \(7!\), and \(8!\), we can follow these steps:

### Step 1: Write the Factorials
We start by expressing the factorials explicitly:
- \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\)
- \(7! = 7 \times 6!\)
- \(8! = 8 \times 7!\)

### Step 2: Identify the H.C.F.
To find the H.C.F., we look for the common factors in all three factorials:
- \(6!\) is present in \(7!\) and \(8!\) as well.
- Therefore, the H.C.F. of \(6!\), \(7!\), and \(8!\) is \(6!\).

### Step 3: Identify the L.C.M.
To find the L.C.M., we can express each factorial in terms of \(6!\):
- \(6! = 6!\)
- \(7! = 7 \times 6!\)
- \(8! = 8 \times 7 \times 6!\)

Now, we can factor out \(6!\):
\[ \text{L.C.M.} = 6! \times \text{L.C.M.}(1, 7, 8) \]
Since \(1\), \(7\), and \(8\) have no common factors, the L.C.M. of these numbers is simply \(7 \times 8\).

### Step 4: Calculate the L.C.M.
Now we calculate:
\[ \text{L.C.M.} = 6! \times (7 \times 8) = 6! \times 56 \]
Since \(6! \times 7 = 7!\) and \(7! \times 8 = 8!\), we can conclude:
\[ \text{L.C.M.} = 8! \]

### Final Results
Thus, we have:
- H.C.F. of \(6!\), \(7!\), and \(8!\) is \(6!\).
- L.C.M. of \(6!\), \(7!\), and \(8!\) is \(8!\).