The least number, which when divided by $48$, $60$, $72$, $108$ and $140$ leaves $38$, $50$, $62$, $98$ and $130$ as remainders respectively is

### Concept & Logic When a number leaves different remainders for different divisors, first find the difference ($K$) between each divisor and its corresponding remainder. If $K$ is constant, the formula is: $$\text{Required Number} = \text{LCM of Divisors} - K$$ ### Step-by-Step Solution * **Check for common difference (K):** $48 - 38 = 10$ $60 - 50 = 10$ $72 - 62 = 10$ $108 - 98 = 10$ $140 - 130 = 10$ The common difference $K = 10$. * **Find the LCM:** $48 = 2^4 \times 3^1$ $60 = 2^2 \times 3^1 \times 5^1$ $72 = 2^3 \times 3^2$ $108 = 2^2 \times 3^3$ $140 = 2^2 \times 5^1 \times 7^1$ Take the highest powers of all prime factors ($2$, $3$, $5$, $7$): $\text{LCM} = 2^4 \times 3^3 \times 5^1 \times 7^1 = 16 \times 27 \times 35 = 15120$. * **Calculate the number:** $$\text{Required Number} = 15120 - 10 = 15110$$ ### Exam Strategy & Shortcut **Option Elimination:** Since $(\text{Number} + 10)$ must be a multiple of the LCM, test the options by adding $10$: (a) $11115 + 10 = 11125$ (Ends in 5, not a multiple of 16/48) (b) $15110 + 10 = 15120$ (Check if $15120$ is divisible by all divisors; yes, it is the LCM). The shortcut is simply identifying the LCM and subtracting the common difference. ### Common Pitfall Adding the common difference instead of subtracting it. Always remember: if remainders are different, you subtract the constant difference $K$ from the LCM. ### Final Answer **Therefore, the correct answer is 15110.**