Find the greatest number which will divide 42, 49 and 56 and leave remainders 6, 7 and 8 respectively

Options

A. 6
B. 8
C. 12
D. 24

Answer

To solve the problem of finding the greatest number that divides 42, 49, and 56 while leaving remainders of 6, 7, and 8 respectively, we can follow these steps:

### Step 1: Adjust the numbers by subtracting the remainders
To find the number that divides the given numbers leaving specific remainders, we first subtract the remainders from each of the numbers.

- For 42, subtract 6:
\( 42 - 6 = 36 \)

- For 49, subtract 7:
\( 49 - 7 = 42 \)

- For 56, subtract 8:
\( 56 - 8 = 48 \)

Now we have the adjusted numbers: 36, 42, and 48.

### Step 2: Find the HCF of the adjusted numbers
Next, we need to find the highest common factor (HCF) of the adjusted numbers: 36, 42, and 48.

**Finding the HCF:**

1. **Prime factorization of each number:**
- 36: \( 2^2 \times 3^2 \)
- 42: \( 2^1 \times 3^1 \times 7^1 \)
- 48: \( 2^4 \times 3^1 \)

2. **Identify the common prime factors:**
- The common prime factors are 2 and 3.

3. **Take the lowest power of each common factor:**
- For 2: The lowest power is \( 2^1 \).
- For 3: The lowest power is \( 3^1 \).

4. **Calculate the HCF:**
\[ HCF = 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Step 3: Conclusion
The greatest number that will divide 42, 49, and 56 and leave remainders of 6, 7, and 8 respectively is **6**.

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