What is the least number which when divided by the numbers 3, 5, 6, 8, 10 and 12 leaves in each case a remainder 2 but when divided by 13 leaves no remainder?

### Concept & Logic To find a number that leaves a constant remainder $R$ when divided by a set of numbers, it must be of the form $(\text{L.C.M.} \times k) + R$, where $k$ is an integer ($1, 2, 3 \dots$). To satisfy an additional divisibility condition (like being perfectly divisible by 13), we must find the smallest value of $k$ that makes the entire expression divisible by that number. $$ \text{Required Number} = (\text{L.C.M.} \times k) + R $$ ### Step-by-Step Solution * **Given:** Divisors are 3, 5, 6, 8, 10, and 12. Constant remainder is 2. The final number must be a multiple of 13. * First, find the L.C.M. of 3, 5, 6, 8, 10, and 12. * The L.C.M. is 120. * The required number must be of the form: $120k + 2$. * We need to find the least value of $k$ such that $(120k + 2)$ is exactly divisible by 13. To make testing easier, break 120 down into a multiple of 13 plus a remainder: * $120k = (117k + 3k)$ *(since 117 is exactly divisible by 13)* * So, $(120k + 2) = 117k + (3k + 2)$ * Since $117k$ is always divisible by 13, we only need to find a $k$ where $(3k + 2)$ is divisible by 13. Test small values for $k$: * If $k = 1$: 3(1) + 2 = 5 (No) * If $k = 2$: 3(2) + 2 = 8 (No) * If $k = 3$: 3(3) + 2 = 11 (No) * If $k = 8$: 3(8) + 2 = 26 (Yes! 26 is divisible by 13). * Substitute $k = 8$ back into the original form: * Required number = 120(8) + 2 = 960 + 2 = 962. ### Exam Strategy & Shortcut Use **Divisibility Rules on Options**. The question demands the answer is exactly divisible by 13. Test the options directly: * 962 / 13 = 74 (Perfectly divisible). * 122 / 13 = 9.38 (Fail). * 242 / 13 = 18.61 (Fail). If multiple options are divisible by 13, subtract 2 from them and check if the result is divisible by 10 (ends in 0) and 12. 962 - 2 = 960, which works perfectly. ### Common Pitfall A standard trap is ignoring the $k$ multiplier and simply calculating L.C.M. + Remainder = 120 + 2 = 122. While 122 satisfies the first condition, it completely fails the secondary condition of being divisible by 13. Always apply the multiplier form for compound conditions. ### Final Answer **Therefore, the correct answer is 962.**