Find the largest number which can divided x,y,z and leaves remainder a,b,c

Answer

To find the largest number that can divide \( x, y, z \) and leaves remainders \( a, b, c \) respectively, we can follow these steps:

### Step 1: Understand the relationship
The problem states that when we divide \( x \) by our desired number \( N \), we get a remainder \( a \). This can be expressed mathematically as:
\[ x = k_1 N + a \]
for some integer \( k_1 \). Similarly, we can express the other two conditions:
\[ y = k_2 N + b \]
\[ z = k_3 N + c \]
for integers \( k_2 \) and \( k_3 \).

### Step 2: Rearrange the equations
From the equations above, we can rearrange them to isolate \( N \):
\[ N = \frac{x - a}{k_1} \]
\[ N = \frac{y - b}{k_2} \]
\[ N = \frac{z - c}{k_3} \]

### Step 3: Define new variables
Let’s define new variables based on the remainders:
- Let \( x' = x - a \)
- Let \( y' = y - b \)
- Let \( z' = z - c \)

Now, we can rewrite our equations as:
\[ N = \frac{x'}{k_1} \]
\[ N = \frac{y'}{k_2} \]
\[ N = \frac{z'}{k_3} \]

### Step 4: Find the largest \( N \)
The largest number \( N \) that can divide \( x', y', z' \) is the greatest common divisor (GCD) of these three new variables:
\[ N = \text{GCD}(x', y', z') \]

### Step 5: Conclusion
Thus, the largest number that can divide \( x, y, z \) leaving remainders \( a, b, c \) respectively is:
\[ N = \text{GCD}(x - a, y - b, z - c) \]

### Final Answer
The required largest number is \( N = \text{GCD}(x - a, y - b, z - c) \).

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