Two numbers 1439 and 2393, when divided by a three-digit number individually, leave the same remainder. What is the least possible three-digit number satisfying the above condition?

Options

A. 159
B. 263
C. 311
D. 106

Answer

To solve the problem of finding the least possible three-digit number \( n \) such that both 1439 and 2393 leave the same remainder when divided by \( n \), we can follow these steps:

### Step 1: Understand the condition
When two numbers leave the same remainder when divided by a number \( n \), it implies that their difference is divisible by \( n \).

### Step 2: Calculate the difference
We need to find the difference between the two numbers:
\[ 2393 - 1439 = 954 \]

### Step 3: Find the factors of the difference
Next, we need to find the factors of 954, as \( n \) must be a divisor of this difference.

### Step 4: Factorize 954
To factor 954, we can start dividing it by the smallest prime numbers:
1. 954 is even, so divide by 2:
\[ 954 \div 2 = 477 \]
2. Next, we factor 477. The sum of the digits (4 + 7 + 7 = 18) is divisible by 3, so we divide by 3:
\[ 477 \div 3 = 159 \]
3. Now, we factor 159. Again, the sum of the digits (1 + 5 + 9 = 15) is divisible by 3:
\[ 159 \div 3 = 53 \]
4. Finally, 53 is a prime number.

So, the prime factorization of 954 is:
\[ 954 = 2 \times 3^2 \times 53 \]

### Step 5: List the factors of 954
Now we can find the factors of 954:
- \( 1 \)
- \( 2 \)
- \( 3 \)
- \( 6 \)
- \( 9 \)
- \( 18 \)
- \( 53 \)
- \( 106 \)
- \( 159 \)
- \( 318 \)
- \( 477 \)
- \( 954 \)

### Step 6: Identify the three-digit factors
From the list of factors, the three-digit factors are:
- \( 106 \)
- \( 159 \)
- \( 318 \)
- \( 477 \)
- \( 954 \)

### Step 7: Find the least three-digit factor
The least three-digit factor from the list is:
\[ \text{Least three-digit factor} = 106 \]

### Conclusion
Thus, the least possible three-digit number \( n \) that satisfies the condition is:
\[ \boxed{106} \]