If the seven-figure number $30X0103$ is a multiple of 13, then $X$ is

## Concept & Strategy Use modulo arithmetic based on place values, or apply the block-of-three alternating sum rule for 13 to isolate the unknown digit $X$. ## Step-by-Step Solution **Given:** The number $30X0103$ is perfectly divisible by 13. **Calculation / Deduction:** * **Step 1: Break the number into manageable parts based on place value.** $30X0103 = 3000103 + (X \times 10000)$ * **Step 2: Find the remainder when $3000103$ is divided by $13$.** $3000103 \div 13 = 230777$ with a remainder of $2$. * **Step 3: Find the remainder when $10000$ is divided by $13$.** $10000 \div 13 = 769$ with a remainder of $3$. * **Step 4: Formulate the modulo equation.** For the whole number to be divisible by 13, the sum of their remainders must be a multiple of $13$. Therefore, $(2 + 3X)$ must be a multiple of $13$. * **Step 5: Test the given options for $X$.** * If $X = 1$: $2 + 3(1) = 5$ (Not divisible) * If $X = 6$: $2 + 3(6) = 20$ (Not divisible) * If $X = 7$: $2 + 3(7) = 23$ (Not divisible) * If $X = 8$: $2 + 3(8) = 26$ (Yes, $26$ is a multiple of 13) ## Exam Strategy & Shortcut An alternative faster trick for divisibility by 7, 11, and 13 is taking the alternating sum of blocks of 3 digits from right to left. Blocks: `103`, `0X0`, `3`. Alternating sum: $103 - (0X0) + 3 = 106 - 10X$. Test the options directly into $(106 - 10X)$ to see which yields a multiple of 13. For $X=8 \implies 106 - 80 = 26$. Divides perfectly! ## Common Pitfall When using the block method, a frequent mistake is misinterpreting the place value of the unknown digit within its block. In `0X0`, the $X$ represents the tens place, so its numerical value is $10X$, not just $X$. Writing $103 - X + 3$ will lead to an incorrect answer. ## Final Answer **Therefore, the correct answer is 8.**