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: Write the expression.
The given expression is $17 \times 11 \times 13 + 11$.
Step 2: Factor out the common term.
Notice that 11 is a common factor in both terms: \[ 17 \times 11 \times 13 + 11 = 11(17 \times 13 + 1) \]
Step 3: Simplify the bracket.
\[ 17 \times 13 + 1 = 221 + 1 = 222 \] So the expression becomes: $11 \times 222$.
Step 4: Check if 222 has factors.
$222 = 2 \times 111 = 2 \times 3 \times 37$. So $11 \times 222 = 11 \times 2 \times 3 \times 37$.
Step 5: Determine the nature of the number.
Since the number has more than two factors (1, 2, 3, 11, 37, ...), it is a composite number. It is NOT a prime number.
Step 6: Conclusion.
$17 \times 11 \times 13 + 11$ is a composite number.
\[ \boxed{\text{a composite number}} \]