The greatest number, that divides 122 and 243 leaving respectively 2 and 3 as remainders, is

Options

A. 0.12
B. 24
C. 30
D. 120

Answer

To solve the problem of finding the greatest number that divides 122 and 243 leaving remainders of 2 and 3 respectively, we can follow these steps:

### Step 1: Set up the equations based on the problem
Let the greatest number be \( n \). According to the problem:
- When 122 is divided by \( n \), the remainder is 2.
- When 243 is divided by \( n \), the remainder is 3.

This can be expressed mathematically as:
- \( 122 \equiv 2 \mod n \)
- \( 243 \equiv 3 \mod n \)

### Step 2: Rewrite the equations
From the equations above, we can rewrite them as:
- \( 122 - 2 = 120 \) is divisible by \( n \).
- \( 243 - 3 = 240 \) is divisible by \( n \).

This means that \( n \) must divide both 120 and 240.

### Step 3: Find the GCD of 120 and 240
To find the greatest number \( n \), we need to calculate the greatest common divisor (GCD) of 120 and 240.

1. **Prime Factorization**:
- \( 120 = 2^3 \times 3^1 \times 5^1 \)
- \( 240 = 2^4 \times 3^1 \times 5^1 \)

2. **GCD Calculation**:
- For \( 2 \): The minimum power is \( 2^3 \).
- For \( 3 \): The minimum power is \( 3^1 \).
- For \( 5 \): The minimum power is \( 5^1 \).

Thus, the GCD is:
\[ GCD(120, 240) = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \]

### Step 4: Conclusion
The greatest number \( n \) that divides both 122 and 243 leaving remainders of 2 and 3 respectively is:
\[ \boxed{120} \]