Question

$(3 \times 11 \times 13 + 3)$ is :

• A. a prime number
• B. divisible by $13$
• C. a composite number
• D. an odd number

Hint:

Whenever you see a question of the form $a \times b \times c + c$, immediately factor out $c$:
$c \times (a \times b + 1)$.
Since this expresses the number as a product of two integers greater than 1, it must be a composite number.
This algebraic approach avoids tedious calculations and guarantees a correct result quickly.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question is from the chapter "Real Numbers", specifically addressing the classification of numbers as prime or composite.
A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.
A composite number is a natural number greater than 1 that has more than two distinct positive divisors.
We need to analyze the properties of the given numerical expression $(3 \times 11 \times 13 + 3)$ and classify it accordingly.

Step 2: Key Formula or Approach:
To check whether a number of the form $a \cdot b \cdot c + d$ is prime or composite, we can factor out common terms.
By expressing the number as a product of two integers both greater than 1, we can prove it is composite without necessarily computing the entire large multiplication.
Let's find a common factor between the terms of the expression.

Step 3: Detailed Explanation:

• Write down the given expression:
\[ 3 \times 11 \times 13 + 3 \]

• Notice that there are two terms in this expression: the product $(3 \times 11 \times 13)$ and the number $3$.
Both terms share a common factor of $3$.

• Factor out $3$ from the expression:
\[ 3 \times (11 \times 13 + 1) \]

• Simplify the expression inside the parentheses:
\[ 11 \times 13 = 143 \] Now, add 1 to this product:
\[ 143 + 1 = 144 \]

• Now, rewrite the original expression as a product:
\[ 3 \times 144 \]

• Compute the final value:
\[ 3 \times 144 = 432 \]

• Let us analyze the number $432$:

• Since $432 = 3 \times 144$, it has factors other than 1 and itself (such as 2, 3, 4, 6, etc.). Therefore, it is

not a prime number.

• Is it divisible by $13$? Let's divide $432$ by $13$:
$432 = 13 \times 33 + 3$. Since there is a remainder of 3, it is

not divisible by 13.

• Is it an odd number? No, the last digit is 2, so $432$ is an

even number.

• Since $432$ is greater than 1 and has multiple factors, it is a

composite number.


Step 4: Final Answer:
Therefore, the given number is a composite number, which matches Option (C).