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: Understand the Factorials
First, we need to express the factorials in terms of their definitions:
- \(6! = 720\)
- \(7! = 7 \times 6! = 7 \times 720 = 5040\)
- \(8! = 8 \times 7! = 8 \times 5040 = 40320\)

### Step 2: Find the H.C.F.
To find the H.C.F. of \(6!\), \(7!\), and \(8!\), we can express \(7!\) and \(8!\) in terms of \(6!\):
- \(7! = 7 \times 6!\)
- \(8! = 8 \times 7!\) which can also be written as \(8 \times 7 \times 6!\)

The H.C.F. is the highest number that divides all three factorials. Since \(6!\) is a factor of both \(7!\) and \(8!\), we can conclude:
\[ \text{H.C.F.}(6!, 7!, 8!) = 6! \]

### Step 3: Find the L.C.M.
To find the L.C.M. of \(6!\), \(7!\), and \(8!\), we look for the lowest number that is a multiple of all three. Since \(8!\) includes both \(7!\) and \(6!\) as factors, we can conclude:
\[ \text{L.C.M.}(6!, 7!, 8!) = 8! \]

### Final Result
Thus, we have:
- H.C.F. = \(6!\)
- L.C.M. = \(8!\)