Which of the following numbers is divisible by 3, 7, 9 and 11?

### Concept & Strategy To find a number divisible by multiple prime and composite factors, we do not need to test every single number against every single rule. Instead, apply the easiest, most restrictive divisibility rules first (like $9$ and $11$) to rapidly eliminate incorrect options. Any number divisible by $9$ is inherently divisible by $3$. ### Step-by-Step Solution * **Test 1: Divisibility by 9 (Sum of digits must be a multiple of 9)** * (a) $639 \rightarrow 6+3+9 = 18$ (Divisible) * (b) $2079 \rightarrow 2+0+7+9 = 18$ (Divisible) * (c) $3791 \rightarrow 3+7+9+1 = 20$ (Not divisible - **Eliminate**) * (d) $37911 \rightarrow 3+7+9+1+1 = 21$ (Not divisible - **Eliminate**) * **Test 2: Divisibility by 11 (Alternating sum difference)** We only need to test the remaining options (a) and (b). * (a) $639 \rightarrow$ Odd sum: $9+6=15$. Even sum: $3$. Difference: $15-3 = 12$. (Not divisible - **Eliminate**) * (b) $2079 \rightarrow$ Odd sum: $9+0=9$. Even sum: $7+2=9$. Difference: $9-9 = 0$. (Divisible) By elimination, $2079$ is the only valid number. ### Exam Strategy & Shortcut When a question lists $9$ and $3$, ignore $3$ completely. It is redundant information designed to waste your time. Always start your elimination with the rule of $9$ because calculating the digit sum is incredibly fast. Once reduced to two options, the rule of $11$ acts as a quick tie-breaker. You almost never actually need to use the rule of $7$, which is complex and time-consuming. ### Common Pitfall The biggest trap is trying to find the Least Common Multiple (LCM) of $3, 7, 9, \text{and } 11$ and then seeing which option matches it. Calculating $9 \times 7 \times 11 = 693$ and then looking for multiples of $693$ takes much longer than simply using the elimination method with basic digit rules. ### Final Answer Therefore, the correct answer is **2079**.