M is the largest three digit number which when divided by 6 and 5 leaves remainders 5 and 3 respectively. What will be the remainder when M is divided by 11?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the largest three-digit number \( M \) that satisfies the conditions given in the question. Let's break down the steps:

### Step 1: Understand the conditions
We know that:
- When \( M \) is divided by 6, it leaves a remainder of 5.
- When \( M \) is divided by 5, it leaves a remainder of 3.

### Step 2: Set up the equations
From the conditions, we can express \( M \) in terms of the divisors:
1. \( M \equiv 5 \mod 6 \)
2. \( M \equiv 3 \mod 5 \)

### Step 3: Find a common solution
To find a common solution, we can use the method of successive substitutions or the Chinese Remainder Theorem.

First, we can express \( M \) in terms of one of the moduli. Let's express \( M \) in terms of the modulus 5:
- From \( M \equiv 3 \mod 5 \), we can write:
\[ M = 5k + 3 \quad \text{for some integer } k \]

Now substitute this expression into the first condition:
\[ 5k + 3 \equiv 5 \mod 6 \]
Subtract 3 from both sides:
\[ 5k \equiv 2 \mod 6 \]
To solve for \( k \), we can find the multiplicative inverse of 5 modulo 6. Since \( 5 \equiv -1 \mod 6 \), we can rewrite the equation as:
\[ -k \equiv 2 \mod 6 \implies k \equiv -2 \mod 6 \implies k \equiv 4 \mod 6 \]
Thus, we can express \( k \) as:
\[ k = 6m + 4 \quad \text{for some integer } m \]

### Step 4: Substitute back to find \( M \)
Now substitute \( k \) back into the equation for \( M \):
\[ M = 5(6m + 4) + 3 = 30m + 20 + 3 = 30m + 23 \]

### Step 5: Find the largest three-digit number
To find the largest three-digit number \( M \), we need \( M \leq 999 \):
\[ 30m + 23 \leq 999 \]
Subtract 23 from both sides:
\[ 30m \leq 976 \]
Divide by 30:
\[ m \leq \frac{976}{30} \approx 32.53 \]
Thus, the maximum integer value for \( m \) is 32.

### Step 6: Calculate \( M \)
Now substitute \( m = 32 \) back into the equation for \( M \):
\[ M = 30(32) + 23 = 960 + 23 = 983 \]

### Step 7: Find the remainder when \( M \) is divided by 11
Now we need to find the remainder when \( M = 983 \) is divided by 11:
\[ 983 \div 11 = 89 \quad \text{(since } 89 \times 11 = 979\text{)} \]
Now, subtract:
\[ 983 - 979 = 4 \]
Thus, the remainder when \( M \) is divided by 11 is 4.

### Final Answer
The remainder when \( M \) is divided by 11 is \( \boxed{4} \).