Is `7times6times5times4times3times2times1``+5` composite number ? Justify your answer.

Answer

To determine whether the expression \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \) is a composite number, we will follow these steps:

### Step 1: Calculate the factorial
First, we need to calculate \( 7! \) (which is \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)).

\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

Calculating this step by step:
- \( 7 \times 6 = 42 \)
- \( 42 \times 5 = 210 \)
- \( 210 \times 4 = 840 \)
- \( 840 \times 3 = 2520 \)
- \( 2520 \times 2 = 5040 \)
- \( 5040 \times 1 = 5040 \)

Thus, \( 7! = 5040 \).

### Step 2: Add 5 to the factorial
Now, we add 5 to the result from Step 1:

\[ 5040 + 5 = 5045 \]

### Step 3: Check if 5045 is a composite number
A composite number is defined as a number that has at least one factor other than 1 and itself. To check if \( 5045 \) is composite, we need to find its factors.

### Step 4: Factor 5045
We can start by checking divisibility by small prime numbers:

- **Divisibility by 5**: Since \( 5045 \) ends in 5, it is divisible by 5.
\[ 5045 \div 5 = 1009 \]

Thus, we can express \( 5045 \) as:
\[ 5045 = 5 \times 1009 \]

### Step 5: Check if 1009 is prime
Next, we need to check if \( 1009 \) is a prime number. We can do this by checking divisibility by prime numbers up to \( \sqrt{1009} \) (which is approximately 31.76). We check divisibility by the primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31.

- **Divisibility by 2**: \( 1009 \) is odd, not divisible.
- **Divisibility by 3**: The sum of digits \( 1 + 0 + 0 + 9 = 10 \) is not divisible by 3.
- **Divisibility by 5**: \( 1009 \) does not end in 0 or 5.
- **Divisibility by 7**: \( 1009 \div 7 \approx 144.14 \) (not an integer).
- **Divisibility by 11**: \( 1009 \div 11 \approx 91.73 \) (not an integer).
- **Divisibility by 13**: \( 1009 \div 13 \approx 77.61 \) (not an integer).
- **Divisibility by 17**: \( 1009 \div 17 \approx 59.35 \) (not an integer).
- **Divisibility by 19**: \( 1009 \div 19 \approx 53.11 \) (not an integer).
- **Divisibility by 23**: \( 1009 \div 23 \approx 43.87 \) (not an integer).
- **Divisibility by 29**: \( 1009 \div 29 \approx 34.79 \) (not an integer).
- **Divisibility by 31**: \( 1009 \div 31 \approx 32.58 \) (not an integer).

Since \( 1009 \) is not divisible by any of these primes, it is a prime number.

### Conclusion
Since \( 5045 \) can be factored into \( 5 \times 1009 \), and both \( 5 \) and \( 1009 \) are prime, \( 5045 \) has factors other than 1 and itself. Therefore, \( 5045 \) is a composite number.

Thus, we conclude that:

\[ \text{The number } 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \text{ is a composite number.} \]