The least multiple of $13$, which on dividing by $4$, $5$, $6$, $7$ and $8$ leaves remainder $2$ in each case is

### Concept & Logic This is a dual-condition problem. The number must simultaneously be: 1. Of the format: $k \times \text{LCM} + R$ (to satisfy the remainder condition). 2. Perfectly divisible by a secondary target (in this case, $13$). The core equation is: $$\text{Number} = k \times \text{LCM}(a, b, c, \dots) + R$$ We must find the smallest positive integer $k$ ($1, 2, 3, \dots$) that makes the entire expression a perfect multiple of $13$. ### Step-by-Step Solution * **Given:** Divisors are $4$, $5$, $6$, $7$, $8$. Remainder $R = 2$. Target multiple = $13$. * **Find the LCM:** Prime factors: $4 = 2^2$ $5 = 5^1$ $6 = 2^1 \times 3^1$ $7 = 7^1$ $8 = 2^3$ $$LCM = 2^3 \times 3^1 \times 5^1 \times 7^1 = 8 \times 3 \times 5 \times 7 = 840$$ * **Formulate the number:** The number takes the form: $N = 840k + 2$. * **Satisfy the multiple of 13 condition:** We need $(840k + 2)$ to be perfectly divisible by $13$. To simplify testing, divide $840$ by $13$ and find the remainder. $840 \div 13 = 64$ with a remainder of $8$. Rewrite the expression using this remainder: $840k + 2 = (13 \times 64k) + 8k + 2$ Since $(13 \times 64k)$ is already divisible by $13$, we only need to find a $k$ where the remaining part $(8k + 2)$ is divisible by $13$. Test values of $k$: If $k = 1$: $8(1) + 2 = 10$ (Not divisible by 13) If $k = 2$: $8(2) + 2 = 18$ (Not divisible by 13) If $k = 3$: $8(3) + 2 = 26$ (Perfectly divisible by 13, since $13 \times 2 = 26$) * **Calculate final number:** Substitute $k = 3$ back into the original formula: $$N = 840(3) + 2 = 2520 + 2 = 2522$$ ### Exam Strategy & Shortcut **Direct Multiple Verification:** Since the question asks for a "multiple of 13", the correct option MUST be perfectly divisible by $13$. Let's check the options directly: (a) $840 \div 13 = 64.6$ (Not a multiple) (b) $842 \div 13 = 64.7$ (Not a multiple) (c) $2520 \div 13 = 193.8$ (Not a multiple) (d) $2522 \div 13 = 194$ (Exact multiple) Only option (d) satisfies the core condition of being a multiple of $13$. You can solve this in 10 seconds without calculating any LCMs! ### Common Pitfall Students often stop at $k=1$ and calculate $840(1) + 2 = 842$, choosing it if it's an option. You must always ensure the secondary condition (divisible by 13) is met by checking your final number before finalizing the answer. ### Final Answer **Therefore, the correct answer is 2522.**