Explain why: `1xx2xx3xx5xx7+3xx7` is a composite number.

Answer

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

### Step 1: Simplify the expression
First, we need to simplify the expression \( 1 \times 2 \times 3 \times 5 \times 7 + 3 \times 7 \).

### Step 2: Calculate \( 1 \times 2 \times 3 \times 5 \times 7 \)
Calculating the first part:
\[ 1 \times 2 = 2 \]
\[ 2 \times 3 = 6 \]
\[ 6 \times 5 = 30 \]
\[ 30 \times 7 = 210 \]
So, \( 1 \times 2 \times 3 \times 5 \times 7 = 210 \).

### Step 3: Calculate \( 3 \times 7 \)
Now calculating the second part:
\[ 3 \times 7 = 21 \]

### Step 4: Add the two results together
Now we can add the two results:
\[ 210 + 21 = 231 \]

### Step 5: Determine if 231 is a composite number
A composite number is defined as a number that has more than two distinct positive divisors (factors). To check if 231 is composite, we can find its factors.

### Step 6: Factor 231
We can start by checking divisibility by small prime numbers:
- 231 is odd, so it is not divisible by 2.
- The sum of the digits \( 2 + 3 + 1 = 6 \) is divisible by 3, so 231 is divisible by 3.
\[ 231 \div 3 = 77 \]

Now we can factor 77:
- 77 is not divisible by 2 or 3.
- It is divisible by 7:
\[ 77 \div 7 = 11 \]

### Step 7: Write the complete factorization
Thus, we can write:
\[ 231 = 3 \times 7 \times 11 \]

### Conclusion
Since 231 can be factored into three prime numbers (3, 7, and 11), it has more than two factors (1, 3, 7, 11, 21, 33, 77, and 231). Therefore, 231 is a composite number.