A number is multiplied by 11 and 11 is added to the product. If the resulting number is divisible by 13, the smallest original number is

### Concept & Strategy This problem combines basic linear algebra with modular arithmetic (divisibility). Factoring out common multipliers is the fastest way to simplify the divisibility check. $$ 11x + 11 \equiv 0 \pmod{13} $$ ### Step-by-Step Solution * **Step 1: Translate the word problem into algebra.** Let the original unknown number be $x$. * Multiply by 11: $11x$ * Add 11 to the product: $11x + 11$ * **Step 2: Factor the expression.** * $11x + 11 = 11(x + 1)$ * **Step 3: Apply the divisibility condition.** We are given that $11(x + 1)$ is perfectly divisible by 13. * **Step 4: Use co-prime logic.** 11 and 13 are prime numbers; they share no common factors. Therefore, for the entire product $11(x + 1)$ to be divisible by 13, the $(x + 1)$ portion *must* be the part that is divisible by 13. * **Step 5: Find the minimum value.** For $(x + 1)$ to be a multiple of 13, the smallest positive integer multiple is 13 itself. * $x + 1 = 13$ * $x = 12$ ### Exam Strategy & Shortcut **Backsolving from Options:** When asked for the "smallest" number, immediately start plugging the provided options into the described process, starting from the smallest option (a). Test Option (a) 12: * Multiply by 11: $12 \times 11 = 132$ * Add 11: $132 + 11 = 143$ * Check divisibility by 13: $143 \div 13 = 11$. It works perfectly! Since it's the smallest option provided, you don't even need to test the others. ### Common Pitfall A common mistake is forgetting to factor the $11x + 11$ and trying to solve $11x \equiv -11 \pmod{13}$ using complex modular inverses. While mathematically valid, it takes significantly more time than simple factoring or backsolving. ### Final Answer Therefore, the correct answer is **12**.