When 58123 and 59059 are divided by a three digit number N, the same remainder is obtained. How many values can N take?

Options

A. 2
B. 3
C. 6
D. 7

Answer

To solve the problem, we need to find how many three-digit numbers \( N \) can divide the difference between the two numbers \( 58123 \) and \( 59059 \) while yielding the same remainder. Here are the steps to find the solution:

### Step 1: Calculate the Difference
First, we need to find the difference between the two numbers:

\[ 59059 - 58123 = 936 \]

### Step 2: Find Factors of the Difference
Next, we need to find the factors of \( 936 \). A factor of \( 936 \) is any number \( N \) such that \( 936 \div N \) results in an integer.

### Step 3: Factorization of 936
To find the factors, we can perform prime factorization of \( 936 \):

1. Divide by \( 2 \):
\[ 936 \div 2 = 468 \]
2. Divide by \( 2 \) again:
\[ 468 \div 2 = 234 \]
3. Divide by \( 2 \) again:
\[ 234 \div 2 = 117 \]
4. Divide by \( 3 \):
\[ 117 \div 3 = 39 \]
5. Divide by \( 3 \) again:
\[ 39 \div 3 = 13 \]
6. \( 13 \) is a prime number.

So, the prime factorization of \( 936 \) is:
\[ 936 = 2^3 \times 3^2 \times 13^1 \]

### Step 4: Calculate the Number of Factors
To find the total number of factors, we use the formula:
\[ (\text{exponent of } p_1 + 1)(\text{exponent of } p_2 + 1)(\text{exponent of } p_3 + 1) \]
For \( 936 = 2^3 \times 3^2 \times 13^1 \):
- The exponent of \( 2 \) is \( 3 \) (so \( 3 + 1 = 4 \))
- The exponent of \( 3 \) is \( 2 \) (so \( 2 + 1 = 3 \))
- The exponent of \( 13 \) is \( 1 \) (so \( 1 + 1 = 2 \))

Thus, the total number of factors is:
\[ 4 \times 3 \times 2 = 24 \]

### Step 5: Identify Three-Digit Factors
Now we need to find which of these factors are three-digit numbers. We will list the factors of \( 936 \) and check which ones are three-digit numbers.

The factors of \( 936 \) are:
1, 2, 3, 4, 6, 8, 9, 12, 13, 18, 26, 39, 52, 78, 117, 156, 234, 312, 468, 936.

From this list, the three-digit factors are:
- 117
- 156
- 234
- 312
- 468
- 936

### Step 6: Count the Three-Digit Factors
Counting the three-digit factors, we find there are \( 6 \) three-digit factors.

### Final Answer
Thus, the number of values that \( N \) can take is:

\[ \boxed{6} \]