A number less than $500$, when divided by $4$, $5$, $6$, $7$ leaves remainder $1$ in each case. The number is

### Concept & Logic This problem combines the standard "constant remainder" model with a range limit ("less than 500"). The general format for any number that leaves a constant remainder $R$ when divided by given divisors is: $$\text{Number} = k \times \text{LCM}(a, b, c, \dots) + R$$ where $k$ is a positive integer ($1, 2, 3 \dots$). We must choose $k$ such that the final number fits the specific constraint provided in the question. ### Step-by-Step Solution * **Given:** Divisors are $4$, $5$, $6$, and $7$. Remainder $R = 1$. The target number must be $< 500$. * **Find the LCM:** The divisors are small, let's list their prime factors: $4 = 2^2$ $5 = 5^1$ $6 = 2^1 \times 3^1$ $7 = 7^1$ $$LCM = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420$$ * **Formulate the general equation:** The required number format is: $$N = 420k + 1$$ * **Apply the constraint:** We need $N < 500$. Let $k = 1$: $N = 420(1) + 1 = 421$. (This is less than $500$). Let $k = 2$: $N = 420(2) + 1 = 841$. (This exceeds $500$). So the only valid number fitting the constraint is $421$. ### Exam Strategy & Shortcut **Unit Digit Check & Direct Elimination:** The number leaves a remainder of $1$ when divided by $5$. This means the unit digit of the correct option must be either $1$ or $6$. Options available: (a) 211 (b) 420 (Eliminated) (c) 421 (d) 441 Now, the number must also leave a remainder of $1$ when divided by $4$. This means $(Option - 1)$ must be perfectly divisible by $4$. (a) $211 - 1 = 210$. ($210$ is not divisible by $4$). Eliminated. (c) $421 - 1 = 420$. ($420$ is divisible by $4$). Valid. (d) $441 - 1 = 440$. ($440$ is divisible by $4$). Valid. Finally, check divisibility by $7$ for the remaining options. $(Option - 1)$ must be divisible by $7$. $420 \div 7 = 60$. (Perfectly divisible). Option (c) is correct! ### Common Pitfall Students often just calculate the LCM ($420$) and select it as the answer, forgetting that $420$ leaves a remainder of $0$, not $1$. Always remember to add the constant remainder back to the LCM multiple. ### Final Answer **Therefore, the correct answer is 421.**