Find the largest three digit number which leave remainder as one being divided by 2, 5, 9 & 11 in each case ?

Options

A. 991
B. 997
C. 981
D. 983

Answer

To find the largest three-digit number that leaves a remainder of 1 when divided by 2, 5, 9, and 11, we can follow these steps:

### Step 1: Identify the largest three-digit number
The largest three-digit number is 999.

### Step 2: Find the LCM of the divisors
Since 2, 5, 9, and 11 are co-prime (they have no common factors), we can find the LCM by multiplying them together:
\[ \text{LCM} = 2 \times 5 \times 9 \times 11 \]

### Step 3: Calculate the LCM
Calculating the LCM:
\[ 2 \times 5 = 10 \]
\[ 10 \times 9 = 90 \]
\[ 90 \times 11 = 990 \]
Thus, the LCM of 2, 5, 9, and 11 is 990.

### Step 4: Find the largest multiple of LCM less than or equal to 999
Now, we need to find the largest multiple of 990 that is less than or equal to 999. Since 990 is less than 999, we can use it directly.

### Step 5: Adjust for the remainder
We need the number that leaves a remainder of 1 when divided by 2, 5, 9, and 11. Therefore, we take the largest multiple of 990 and add 1:
\[ 990 + 1 = 991 \]

### Conclusion
The largest three-digit number that leaves a remainder of 1 when divided by 2, 5, 9, and 11 is **991**.

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