Question

\(17 \times 11 \times 13 + 11\) is

• A. a prime number.
• B. multiple of 17.
• C. a composite number.
• D. an odd number.

Hint:

Whenever you have an expression of the form \(a \cdot b \cdot c + b\), you can factor out \(b\) to get \(b(ac + 1)\).
Since \(b \gt 1\) and \(ac + 1 \gt 1\), this product represents a number with factors other than 1 and itself, making it automatically composite!
This allows you to answer the question without carrying out any long multiplication.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given an arithmetic expression: \(17 \times 11 \times 13 + 11\).
We need to analyze this expression to classify it as prime, multiple of 17, composite, or odd.

Step 2: Key Formula or Approach:
1. A prime number is a number greater than 1 that has only two factors: 1 and itself.
2. A composite number has at least one divisor other than 1 and itself.
We can check for common factors by factoring out common terms in the given expression algebraically.

Step 3: Detailed Explanation:

• Let us write the given expression:
\[ N = 17 \times 11 \times 13 + 11 \]

• Notice that the number 11 is common to both terms in the expression.
- Let us factor out 11:
\[ N = 11 \times (17 \times 13 + 1) \]

• Evaluate the expression inside the parentheses:
- Calculate the product \(17 \times 13\):
\[ 17 \times 13 = 221 \] - Add 1 to the product:
\[ 17 \times 13 + 1 = 221 + 1 = 222 \]

• Therefore, the original number \(N\) can be written as:
\[ N = 11 \times 222 \] - This can be factored further since 222 is an even number:
\[ N = 11 \times 2 \times 111 = 22 \times 111 = 2442 \]

• Let us analyze the factors of \(N = 2442\):
- Since \(2442\) is divisible by 2, 11, and other integers, it has multiple prime factors.
- Therefore, \(2442\) is a composite number.
- Since it is divisible by 2, it is an even number, not an odd number.
- Also, 17 does not divide 2442 evenly (since \(2442 = 17 \times 143 + 11\)), so it is not a multiple of 17.


Step 4: Final Answer:
The given number is a composite number. This matches option (C).