What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively ?

Options

A. 13
B. 9
C. 3
D. 585

Explanation

To solve the problem of finding the largest number that divides 70 and 125, leaving remainders of 5 and 8 respectively, we can follow these steps:

### Step 1: Understand the problem
We need to find a number \( x \) such that:
- When 70 is divided by \( x \), the remainder is 5.
- When 125 is divided by \( x \), the remainder is 8.

### Step 2: Set up the equations
From the information given, we can express the conditions mathematically:
- \( 70 - 5 = 65 \) is divisible by \( x \).
- \( 125 - 8 = 117 \) is also divisible by \( x \).

So, we need to find \( x \) such that:
- \( x \) divides 65
- \( x \) divides 117

### Step 3: Find the factors of 65 and 117
Let's find the factors of both numbers.

**Factors of 65:**
- The prime factorization of 65 is \( 5 \times 13 \).
- The factors of 65 are: 1, 5, 13, 65.

**Factors of 117:**
- The prime factorization of 117 is \( 3 \times 39 \) or \( 3 \times 3 \times 13 \).
- The factors of 117 are: 1, 3, 9, 13, 39, 117.

### Step 4: Find the common factors
Now, we need to find the common factors of 65 and 117:
- The common factors are: 1 and 13.

### Step 5: Identify the largest common factor
Among the common factors, the largest one is 13.

### Conclusion
Thus, the largest number that divides 70 and 125, leaving remainders of 5 and 8 respectively, is **13**.