Find the least multiple of $23$, which when divided by $18$, $21$ and $24$ leaves remainders $7$, $10$ and $13$ respectively.

### Concept & Strategy This is a two-step problem: 1. Find a general expression for numbers leaving those specific remainders using the "Constant Difference" rule. 2. Find the smallest value of that expression that is also divisible by $23$. ### Step-by-Step Solution * **Constant Difference ($K$):** $18 - 7 = 11$ $21 - 10 = 11$ $24 - 13 = 11$ Common difference $K = 11$. * **Find LCM of divisors ($18, 21, 24$):** $18 = 2 \times 3^2$ $21 = 3 \times 7$ $24 = 2^3 \times 3$ $\text{LCM} = 2^3 \times 3^2 \times 7 = 8 \times 9 \times 7 = 504$. * **General expression:** Number $N = 504k - 11$. * **Find $k$ such that $N$ is divisible by $23$:** $504 \div 23 = 21$ remainder $21$. $N = (23 \times 21k) + 21k - 11$. We need $(21k - 11)$ to be a multiple of $23$. If $k=1$, $21-11=10$ (No). If $k=2$, $42-11=31$ (No). If $k=3$, $63-11=52$ (No). If $k=6$, $126-11=115$. $115 \div 23 = 5$. (Yes!) $N = 504(6) - 11 = 3024 - 11 = 3013$. ### Exam Strategy & Shortcut **Direct Verification:** The question asks for a multiple of $23$. Check the options: (a) $3002 \div 23 \approx 130.5$ (b) $3013 \div 23 = 131$ (Correct) (c) $3024 \div 23 \approx 131.4$ (d) $3036 \div 23 \approx 132$ Testing divisibility by $23$ is the fastest way to confirm the answer. ### Common Pitfall Trying to solve for $k$ using large numbers. Always reduce the LCM by the divisor ($23$) first to work with smaller remainders. ### Final Answer **Therefore, the correct answer is 3013.**