Find the greatest number that divides 130, 305 and 245 leaving remainders 6, 9 and 17, respectively?

Options

A. 4
B. 5
C. 14
D. 24

Answer

To find the greatest number that divides 130, 305, and 245 leaving remainders 6, 9, and 17 respectively, we can follow these steps:

### Step 1: Adjust the numbers by subtracting the remainders
We need to subtract the remainders from the given numbers to find the actual numbers that the greatest divisor will divide.

- For 130 with a remainder of 6:
\( 130 - 6 = 124 \)

- For 305 with a remainder of 9:
\( 305 - 9 = 296 \)

- For 245 with a remainder of 17:
\( 245 - 17 = 228 \)

So, we have the adjusted numbers: 124, 296, and 228.

### Step 2: Find the HCF (Highest Common Factor) of the adjusted numbers
Next, we need to find the HCF of the numbers 124, 296, and 228.

#### Factorization of 124:
- \( 124 = 2 \times 62 \)
- \( 62 = 2 \times 31 \)
- Thus, \( 124 = 2^2 \times 31 \)

#### Factorization of 296:
- \( 296 = 2 \times 148 \)
- \( 148 = 2 \times 74 \)
- \( 74 = 2 \times 37 \)
- Thus, \( 296 = 2^3 \times 37 \)

#### Factorization of 228:
- \( 228 = 2 \times 114 \)
- \( 114 = 2 \times 57 \)
- \( 57 = 3 \times 19 \)
- Thus, \( 228 = 2^2 \times 3 \times 19 \)

### Step 3: Identify the common factors
Now, we look for the common factors in the factorizations:
- For 124: \( 2^2 \)
- For 296: \( 2^3 \)
- For 228: \( 2^2 \)

The common factor is \( 2^2 \).

### Step 4: Calculate the HCF
The HCF is the product of the lowest powers of the common prime factors:
- The only common prime factor is 2, and the lowest power is \( 2^2 = 4 \).

### Final Answer
The greatest number that divides 130, 305, and 245 leaving remainders 6, 9, and 17 respectively is **4**.

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