Is `7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1 +5` a 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 \( 7! \) (7 factorial), which is the product of all positive integers up to 7.

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

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
Next, we add 5 to the factorial result:

\[ 5040 + 5 = 5045 \]

### Step 3: Determine if 5045 is a composite number
A composite number is defined as a positive integer that has at least one positive divisor other than one and itself. To check if 5045 is composite, we can perform a prime factorization or check for divisibility by prime numbers.

#### Check divisibility:
1. **Divisibility by 2**: 5045 is odd, so it is not divisible by 2.
2. **Divisibility by 3**: The sum of the digits \( 5 + 0 + 4 + 5 = 14 \) is not divisible by 3, so 5045 is not divisible by 3.
3. **Divisibility by 5**: The last digit is 5, so 5045 is divisible by 5.

Calculating \( 5045 \div 5 \):

\[ 5045 \div 5 = 1009 \]

Since 5045 can be expressed as \( 5 \times 1009 \), it has divisors other than 1 and itself.

### Conclusion:
Since 5045 has divisors other than 1 and itself, it is a composite number.

Thus, the answer is:

**Yes, \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \) is a composite number.**

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