Divisibility Rule Explanation

This question tests the understanding of divisibility rules, specifically when a number is divisible by two different numbers.

Understanding Divisibility

Let the number be denoted by $N$. The question states that the number $N$ is divisible by both 11 and 13.

• Being divisible by 11 means $N$ is a multiple of 11. We can write this as $N = 11k$ for some integer $k$.
• Being divisible by 13 means $N$ is a multiple of 13. We can write this as $N = 13m$ for some integer $m$.

Applying the Rule for Coprime Numbers

A fundamental rule in number theory states that if a number $N$ is divisible by two coprime integers $a$ and $b$, then $N$ must also be divisible by their product, $a \times b$.

Two integers are considered coprime if their greatest common divisor (GCD) is 1. Prime numbers, like 11 and 13, are always coprime to each other (unless they are the same prime).

In this case:

• $a = 11$
• $b = 13$
• Since 11 and 13 are prime numbers, they are coprime (GCD(11, 13) = 1).

Therefore, if a number $N$ is divisible by both 11 and 13, it must be divisible by their product $(11 \times 13)$.

Evaluating the Options

Let's analyze the given options based on this principle:

• Option 1: divisible by $(11\times13)$: Since 11 and 13 are coprime, any number divisible by both must be divisible by their product. $11 \times 13 = 143$. So, the number must be divisible by 143. This aligns with our understanding.
• Option 2: divisible by 42: There is no direct rule that connects divisibility by 11 and 13 to divisibility by 42 ($42 = 2 \times 3 \times 7$). A number divisible by 11 and 13 (e.g., 143) is not necessarily divisible by 42.
• Option 3: divisible by $(11+13)$: $11 + 13 = 24$. A number divisible by 11 and 13 is not guaranteed to be divisible by 24. For example, 143 is divisible by 11 and 13, but not by 24.
• Option 4: divisible by $(13-11)$: $13 - 11 = 2$. While numbers divisible by 11 and 13 are often even (like 143*2 = 286), being divisible by 2 is not a necessary consequence of being divisible by 11 and 13 individually. The core rule concerns the product. However, since $11 \times 13 = 143$ (an odd number), the multiples ($143, 286, 429, ...$) alternate between odd and even. So, it's not *always* divisible by 2. More importantly, the rule states divisibility by the product.

Conclusion

Based on the property of divisibility for coprime numbers, if a number is divisible by both 11 and 13, it must be divisible by their product $(11 \times 13)$.