On being divided by a three digit number , the numbers 95336 and 91545 leave the same remainder. Find the number and the remainder

Options

A. 234 and 109
B. 233 and 105
C. 223 and 115
D. None of these

Answer

To solve the problem step by step, we need to find a three-digit number that divides both 95336 and 91545, leaving the same remainder.

### Step 1: Understand the problem
We know that when two numbers leave the same remainder when divided by a certain number, the difference between the two numbers must be divisible by that number.

### Step 2: Calculate the difference
Let's calculate the difference between the two numbers:
\[ 95336 - 91545 = 3791 \]

### Step 3: Find the factors of the difference
Now, we need to find the factors of 3791 to identify the three-digit number. We can start by factoring 3791.

### Step 4: Factor 3791
To factor 3791, we can check for divisibility by prime numbers:
- 3791 is not divisible by 2 (it's odd).
- Check divisibility by 3: \(3 + 7 + 9 + 1 = 20\) (not divisible by 3).
- Check divisibility by 5: last digit is not 0 or 5.
- Check divisibility by 7: \(3791 ÷ 7 \approx 541.57\) (not divisible).
- Check divisibility by 11: \(3 - 7 + 9 - 1 = 4\) (not divisible).
- Check divisibility by 13: \(3791 ÷ 13 \approx 291.615\) (not divisible).
- Check divisibility by 17: \(3791 ÷ 17 = 223\) (divisible).

Thus, we can express:
\[ 3791 = 17 \times 223 \]

### Step 5: Identify the three-digit number
Since 223 is a three-digit number, we can conclude that the three-digit number we are looking for is:
\[ 223 \]

### Step 6: Find the remainder
Next, we need to find the remainder when either of the original numbers is divided by 223. Let's use 95336 for this calculation:
\[ 95336 ÷ 223 \]

Calculating the division:
1. \(223 \times 427 = 95221\)
2. Subtracting gives us the remainder:
\[ 95336 - 95221 = 115 \]

### Final Answer
Thus, the three-digit number is:
\[ \text{Three-digit number} = 223 \]
And the remainder is:
\[ \text{Remainder} = 115 \]