Explain why ` 7 xx 11 xx 13 + 13and 7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1 + 5 ` are composite numbers

Answer

To determine why the numbers \( 7 \times 11 \times 13 + 13 \) and \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \) are composite numbers, we will analyze each expression step by step.

### Step 1: Analyze the first expression \( 7 \times 11 \times 13 + 13 \)

1. **Factor out the common term**:
We can see that both terms in the expression have a common factor of 13.
\[ 7 \times 11 \times 13 + 13 = 13(7 \times 11 + 1) \]

2. **Calculate \( 7 \times 11 + 1 \)**:
Now we compute \( 7 \times 11 \):
\[ 7 \times 11 = 77 \]
Therefore,
\[ 7 \times 11 + 1 = 77 + 1 = 78 \]

3. **Rewrite the expression**:
Now we can rewrite our expression:
\[ 7 \times 11 \times 13 + 13 = 13 \times 78 \]

4. **Check for factors**:
Since 13 is a factor of the expression, and 78 is greater than 1, we can conclude that \( 13 \times 78 \) has factors other than 1 and itself. Hence, it is a composite number.

### Step 2: Analyze the second expression \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \)

1. **Factor out the common term**:
Here, we can factor out 5:
\[ 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 = 5(7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1) \]

2. **Calculate \( 7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1 \)**:
First, compute \( 7 \times 6 \times 4 \times 3 \times 2 \times 1 \):
\[ 7 \times 6 = 42 \]
\[ 42 \times 4 = 168 \]
\[ 168 \times 3 = 504 \]
\[ 504 \times 2 = 1008 \]
\[ 1008 \times 1 = 1008 \]
Now add 1:
\[ 1008 + 1 = 1009 \]

3. **Rewrite the expression**:
Now we can rewrite our expression:
\[ 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 = 5 \times 1009 \]

4. **Check for factors**:
Since 5 is a factor of the expression, and 1009 is greater than 1, we can conclude that \( 5 \times 1009 \) has factors other than 1 and itself. Hence, it is also a composite number.

### Conclusion
Both \( 7 \times 11 \times 13 + 13 \) and \( 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \) are composite numbers because they can be factored into products that include factors other than 1 and themselves.