The four digit smallest positive number which when divided by 4,5,6 or 7, it leaves always the remainder as 3 :

Options

A. 1000
B. 1257
C. 1263
D. 1683

Answer

To find the smallest four-digit positive number that leaves a remainder of 3 when divided by 4, 5, 6, or 7, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( n \) such that:
- \( n \mod 4 = 3 \)
- \( n \mod 5 = 3 \)
- \( n \mod 6 = 3 \)
- \( n \mod 7 = 3 \)

This means that \( n - 3 \) must be divisible by 4, 5, 6, and 7.

### Step 2: Find the Least Common Multiple (LCM)
To find a number that is divisible by 4, 5, 6, and 7, we need to calculate the least common multiple (LCM) of these numbers.

- The prime factorization of each number:
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 7 = 7^1 \)

The LCM is found by taking the highest power of each prime:
- From \( 4 \): \( 2^2 \)
- From \( 5 \): \( 5^1 \)
- From \( 6 \): \( 3^1 \)
- From \( 7 \): \( 7^1 \)

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 \]

Calculating this step by step:
- \( 4 \times 3 = 12 \)
- \( 12 \times 5 = 60 \)
- \( 60 \times 7 = 420 \)

So, \( \text{LCM}(4, 5, 6, 7) = 420 \).

### Step 3: Set Up the Equation
Since \( n - 3 \) must be a multiple of 420, we can express \( n \) as:
\[ n = 420k + 3 \]
where \( k \) is a non-negative integer.

### Step 4: Find the Smallest Four-Digit Number
We need \( n \) to be at least 1000:
\[ 420k + 3 \geq 1000 \]
Subtracting 3 from both sides:
\[ 420k \geq 997 \]
Dividing both sides by 420:
\[ k \geq \frac{997}{420} \approx 2.3738 \]
Since \( k \) must be an integer, we take \( k = 3 \).

### Step 5: Calculate \( n \)
Substituting \( k = 3 \) back into the equation for \( n \):
\[ n = 420 \times 3 + 3 = 1260 + 3 = 1263 \]

### Conclusion
The smallest four-digit positive number which leaves a remainder of 3 when divided by 4, 5, 6, or 7 is:
\[ \boxed{1263} \]