Sure, let's analyze and determine if the number [tex]\( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \)[/tex] is composite.
1. Calculate the factorial part:
First, we need to compute the product [tex]\( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)[/tex].
This is [tex]\( 7! \)[/tex] (7 factorial).
[tex]\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040
\][/tex]
2. Add 5 to the factorial result:
Next, we'll take the result from the previous step and add 5.
[tex]\[
5040 + 5 = 5045
\][/tex]
3. Check if 5045 is composite:
A composite number has factors other than 1 and itself. To verify whether 5045 is composite, we'll need to see if there are any divisors between 1 and 5045.
Let's check for divisibility by smaller prime numbers:
- Divisibility by 2:
5045 is odd, not divisible by 2.
- Divisibility by 3:
Sum of digits in 5045 is [tex]\( 5 + 0 + 4 + 5 = 14 \)[/tex], which is not divisible by 3.
- Divisibility by 5:
The last digit of 5045 is 5, which means 5045 is divisible by 5.
[tex]\[
\frac{5045}{5} = 1009
\][/tex]
We have found that 5045 is divisible by 5 and 1009, confirming 5045 has factors other than 1 and itself.
Therefore, the number [tex]\( 5045 \)[/tex] is composite. Hence, [tex]\( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \)[/tex] is indeed composite.
