Explain why: `7xx11xx13xx15+15` is a composite number.

Answer

To determine why the expression \( 7 \times 11 \times 13 \times 15 + 15 \) is a composite number, we will follow these steps:

### Step 1: Factor out the common term
We notice that \( 15 \) is a common term in both parts of the expression. We can factor it out:

\[ 7 \times 11 \times 13 \times 15 + 15 = 15 \times (7 \times 11 \times 13 + 1) \]

### Step 2: Calculate \( 7 \times 11 \times 13 \)
Now, we need to calculate \( 7 \times 11 \times 13 \):

\[ 7 \times 11 = 77 \]
\[ 77 \times 13 = 1001 \]

### Step 3: Add 1 to the product
Next, we add \( 1 \) to the product we just calculated:

\[ 7 \times 11 \times 13 + 1 = 1001 + 1 = 1002 \]

### Step 4: Rewrite the expression
Now we can rewrite the expression with the calculated values:

\[ 15 \times 1002 \]

### Step 5: Determine the factors of the expression
Now we need to analyze the factors of \( 15 \times 1002 \).

- The number \( 15 \) can be factored as \( 3 \times 5 \).
- The number \( 1002 \) can be factored further. We can find its prime factors:

\[ 1002 = 2 \times 501 \]
\[ 501 = 3 \times 167 \]

So, the complete factorization of \( 1002 \) is:

\[ 1002 = 2 \times 3 \times 167 \]

### Step 6: Combine the factors
Now, combining the factors from both parts:

\[ 15 \times 1002 = (3 \times 5) \times (2 \times 3 \times 167) \]

### Step 7: Identify the total factors
From the factorization, we can see that the expression has the following factors:

- \( 2 \)
- \( 3 \)
- \( 5 \)
- \( 167 \)

Since there are more than two distinct factors, we conclude that \( 15 \times 1002 \) is a composite number.

### Conclusion
Thus, \( 7 \times 11 \times 13 \times 15 + 15 \) is a composite number because it has more than two factors.

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