A number which when divided by 3, 4, 5, 6, & 7 leaves respectively, the remainder 2, 3, 4, 5 and 6. Such smallest 6 digit number is

Options

A. a) 100379
B. b) 1000379
C. c) 100019
D. d) None of these

Answer

To find the smallest six-digit number that, when divided by 3, 4, 5, 6, and 7 leaves remainders of 2, 3, 4, 5, and 6 respectively, we can follow these steps:

### Step 1: Understand the Problem
We need a number \( N \) such that:
- \( N \mod 3 = 2 \)
- \( N \mod 4 = 3 \)
- \( N \mod 5 = 4 \)
- \( N \mod 6 = 5 \)
- \( N \mod 7 = 6 \)

### Step 2: Rewrite the Conditions
We can rewrite these conditions in terms of \( N \):
- \( N = 3k + 2 \)
- \( N = 4m + 3 \)
- \( N = 5n + 4 \)
- \( N = 6p + 5 \)
- \( N = 7q + 6 \)

This means that \( N + 1 \) must be divisible by each of these numbers:
- \( N + 1 \equiv 0 \mod 3 \)
- \( N + 1 \equiv 0 \mod 4 \)
- \( N + 1 \equiv 0 \mod 5 \)
- \( N + 1 \equiv 0 \mod 6 \)
- \( N + 1 \equiv 0 \mod 7 \)

### Step 3: Find the LCM
To find a number that satisfies all these conditions, we need to find the least common multiple (LCM) of the divisors:
- The LCM of \( 3, 4, 5, 6, \) and \( 7 \).

Calculating the LCM:
- The prime factorization gives us:
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 7 = 7^1 \)

The LCM is:
\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420 \]

### Step 4: Set Up the Equation
Now, we know:
\[ N + 1 = 420k \quad \text{for some integer } k \]
Thus,
\[ N = 420k - 1 \]

### Step 5: Find the Smallest Six-Digit Number
We need \( N \) to be a six-digit number:
\[ 100000 \leq 420k - 1 < 1000000 \]
Adding 1 to all parts:
\[ 100001 \leq 420k < 1000001 \]
Dividing by 420:
\[ \frac{100001}{420} \leq k < \frac{1000001}{420} \]
Calculating the bounds:
\[ k \geq \lceil 238.095 \rceil = 239 \]
\[ k < \lfloor 2380.002 \rfloor = 2380 \]

### Step 6: Calculate \( N \) for \( k = 239 \)
Now, we calculate \( N \) for \( k = 239 \):
\[ N = 420 \times 239 - 1 = 100380 - 1 = 100379 \]

### Step 7: Check if \( N \) is Six-Digit
Since \( 100379 \) is a six-digit number, we check if it satisfies the original conditions:
- \( 100379 \mod 3 = 2 \)
- \( 100379 \mod 4 = 3 \)
- \( 100379 \mod 5 = 4 \)
- \( 100379 \mod 6 = 5 \)
- \( 100379 \mod 7 = 6 \)

All conditions are satisfied.

### Final Answer
The smallest six-digit number that meets the conditions is:
\[ \boxed{100379} \]