Explain why (19 x 17 x 11 x 2 + 17 × 19 × 5) is a composite number?

To determine why the expression \( 19 \times 17 \times 11 \times 2 + 17 \times 19 \times 5 \) results in a composite number, let's break it down step by step.

### Step 1: Factor the expression
Notice that the first term \( 19 \times 17 \times 11 \times 2 \) and the second term \( 19 \times 17 \times 5 \) have common factors. We can factor out the common terms \( 19 \times 17 \) from the expression:

\[
19 \times 17 \times (11 \times 2 + 5)
\]

### Step 2: Simplify the expression inside the parentheses
Now, calculate the expression inside the parentheses:

\[
11 \times 2 + 5 = 22 + 5 = 27
\]

So the expression becomes:

\[
19 \times 17 \times 27
\]

### Step 3: Analyze the product
The expression \( 19 \times 17 \times 27 \) is a product of three numbers: 19, 17, and 27.

- 19 is a prime number.
- 17 is a prime number.
- 27 is not a prime number (it's \( 3^3 \)).

Since this expression is a product of three numbers, where two are prime and one is composite (27), the entire product will be a composite number. 

A **composite number** is defined as a positive integer that has at least one divisor other than 1 and itself. In this case, the product \( 19 \times 17 \times 27 \) has divisors other than 1 and itself (specifically, 19, 17, and 27), making it a composite number.

Would you like more details on any part of this explanation? Here are some related questions:

1. What is the difference between a prime number and a composite number?
2. How do you factorize large numbers efficiently?
3. What are the properties of prime numbers?
4. Can every composite number be expressed as a product of prime numbers?
5. What is the significance of the number 27 being \( 3^3 \)?

**Tip:** When simplifying complex expressions, always look for common factors to make the problem easier to solve.

Understand why the expression (19 x 17 x 11 x 2 + 17 × 19 × 5) results in a composite number through a step-by-step breakdown. Explore prime and composite numbers, factorization, and mathematical properties. Suitable for students in Grades 7-9.

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