M is the largest 4 digit number which when divided by 4, 5, 6 and 7 leaves remainder as 2, 3, 4, and 5 respectively . What will be the remainder when M is divided by 9 ?

Options

A. 2
B. 1
C. 3
D. 6

Answer

To solve the problem step by step, we need to find the largest four-digit number \( M \) that leaves specific remainders when divided by 4, 5, 6, and 7. Here’s how we can approach it:

### Step 1: Understand the Remainders
We know that:
- When \( M \) is divided by 4, the remainder is 2.
- When \( M \) is divided by 5, the remainder is 3.
- When \( M \) is divided by 6, the remainder is 4.
- When \( M \) is divided by 7, the remainder is 5.

### Step 2: Set Up the Equations
From the information above, we can express \( M \) in terms of these divisors:
- \( M \equiv 2 \mod 4 \)
- \( M \equiv 3 \mod 5 \)
- \( M \equiv 4 \mod 6 \)
- \( M \equiv 5 \mod 7 \)

### Step 3: Find the LCM
To solve these congruences, we first find the least common multiple (LCM) of the divisors (4, 5, 6, and 7):
- The prime factorization gives us:
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 7 = 7^1 \)

The LCM is calculated as:
\[ \text{LCM}(4, 5, 6, 7) = 2^2 \times 3^1 \times 5^1 \times 7^1 = 420 \]

### Step 4: Find the Largest 4-Digit Number
The largest four-digit number is 9999. We need to find the largest number less than or equal to 9999 that satisfies the conditions above.

### Step 5: Calculate the Nearest Multiple of LCM
To find the largest number \( M \) that is less than or equal to 9999 and is a multiple of 420:
1. Divide 9999 by 420:
\[ 9999 \div 420 \approx 23.8 \]
2. Take the integer part (23) and multiply by 420:
\[ 23 \times 420 = 9660 \]

### Step 6: Adjust for Remainders
Now, we need to adjust 9660 to meet the specific remainder conditions:
- Since \( M \equiv 2 \mod 4 \), we check:
\[ 9660 \mod 4 = 0 \quad (\text{not } 2) \]
- We need to subtract 2:
\[ M = 9660 - 2 = 9658 \]

### Step 7: Verify the Conditions
Now, we can verify:
- \( 9658 \mod 4 = 2 \) (correct)
- \( 9658 \mod 5 = 3 \) (correct)
- \( 9658 \mod 6 = 4 \) (correct)
- \( 9658 \mod 7 = 5 \) (correct)

All conditions are satisfied.

### Step 8: Find the Remainder when Divided by 9
Finally, we need to find the remainder when \( M = 9658 \) is divided by 9:
\[ 9658 \div 9 = 1073 \quad \text{remainder } 1 \]

### Final Answer
The remainder when \( M \) is divided by 9 is **1**.

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