Explain why   `3 xx 5 xx 7 xx 9 xx 11 +11`  is a composite number.

Answer

To determine why the expression \(3 \times 5 \times 7 \times 9 \times 11 + 11\) is a composite number, we can follow these steps:

### Step 1: Calculate the expression
First, we simplify the expression \(3 \times 5 \times 7 \times 9 \times 11 + 11\).

### Step 2: Factor out the common term
Notice that \(11\) is a common term in both parts of the expression:
\[ 3 \times 5 \times 7 \times 9 \times 11 + 11 = 11 \times (3 \times 5 \times 7 \times 9 + 1) \]

### Step 3: Analyze the factors
Now, we need to analyze the expression \(3 \times 5 \times 7 \times 9 + 1\). This expression is an integer, and when we add \(1\) to it, we are still left with an integer.

### Step 4: Identify the factors of the entire expression
Since we factored out \(11\), we can see that the entire expression \(11 \times (3 \times 5 \times 7 \times 9 + 1)\) has at least two factors: \(11\) and \(3 \times 5 \times 7 \times 9 + 1\).

### Step 5: Conclusion about composite numbers
A composite number is defined as a number that has more than two distinct positive factors. Since our expression has at least three factors: \(1\), \(11\), and \(3 \times 5 \times 7 \times 9 + 1\), it meets the criteria for being a composite number.

Thus, we conclude that \(3 \times 5 \times 7 \times 9 \times 11 + 11\) is a composite number.