Find the greatest number of six digits which on being divided by 6,7,8,9 and 10 leaves 4,5,6,7 and 8 in remainder respectively.

Options

A. 997920
B. 997918
C. 997922
D. 997930

Answer

To find the greatest six-digit number that leaves specific remainders when divided by 6, 7, 8, 9, and 10, we can follow these steps:

### Step 1: Understand the Remainders
The problem states that:
- When divided by 6, the remainder is 4.
- When divided by 7, the remainder is 5.
- When divided by 8, the remainder is 6.
- When divided by 9, the remainder is 7.
- When divided by 10, the remainder is 8.

We can express these conditions mathematically:
- \( x \equiv 4 \mod 6 \)
- \( x \equiv 5 \mod 7 \)
- \( x \equiv 6 \mod 8 \)
- \( x \equiv 7 \mod 9 \)
- \( x \equiv 8 \mod 10 \)

### Step 2: Rewrite the Congruences
We can rewrite these congruences in a more uniform way:
- \( x \equiv -2 \mod 6 \)
- \( x \equiv -2 \mod 7 \)
- \( x \equiv -2 \mod 8 \)
- \( x \equiv -2 \mod 9 \)
- \( x \equiv -2 \mod 10 \)

This means that \( x + 2 \) is divisible by 6, 7, 8, 9, and 10.

### Step 3: Find the Least Common Multiple (LCM)
Now, we need to find the LCM of the divisors:
- The prime factorization of the divisors is:
- \( 6 = 2 \times 3 \)
- \( 7 = 7 \)
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)
- \( 10 = 2 \times 5 \)

The LCM will take the highest power of each prime:
- \( LCM = 2^3 \times 3^2 \times 5^1 \times 7^1 = 2520 \)

### Step 4: Find the Greatest Six-Digit Number
The greatest six-digit number is 999999. We need to find the largest number less than or equal to 999999 that satisfies \( x + 2 \equiv 0 \mod 2520 \).

### Step 5: Calculate the Maximum Value
To find this number, we can calculate:
1. Divide 999999 by 2520:
\[ 999999 \div 2520 \approx 396.825 \]
The largest integer less than or equal to this is 396.

2. Multiply back to find the largest multiple of 2520:
\[ 396 \times 2520 = 999120 \]

3. Now, we need to subtract 2 to find \( x \):
\[ x = 999120 - 2 = 999118 \]

### Step 6: Conclusion
Thus, the greatest six-digit number that leaves the specified remainders when divided by 6, 7, 8, 9, and 10 is:
\[ \boxed{999118} \]