Find the greatest number that will divide 445, 572 and 699 , leaving remainders 4, 5, 6, respectively.

Options

A. 42
B. 49
C. 70
D. 63

Answer

To find the greatest number that will divide 445, 572, and 699, leaving remainders 4, 5, and 6 respectively, we can follow these steps:

### Step 1: Adjust the numbers by subtracting the remainders
To find a common divisor, we first need to adjust the numbers by subtracting the respective remainders.

- For 445, subtract 4:
\( 445 - 4 = 441 \)

- For 572, subtract 5:
\( 572 - 5 = 567 \)

- For 699, subtract 6:
\( 699 - 6 = 693 \)

### Step 2: Find the highest common factor (HCF)
Now we need to find the HCF of the adjusted numbers: 441, 567, and 693.

#### Step 2.1: Prime factorization of 441
To find the prime factors of 441:
- 441 can be divided by 3:
\( 441 \div 3 = 147 \)
- 147 can also be divided by 3:
\( 147 \div 3 = 49 \)
- 49 can be divided by 7:
\( 49 \div 7 = 7 \)
- 7 is a prime number.

So, the prime factorization of 441 is:
\( 3^2 \times 7^2 \)

#### Step 2.2: Prime factorization of 567
To find the prime factors of 567:
- 567 can be divided by 3:
\( 567 \div 3 = 189 \)
- 189 can also be divided by 3:
\( 189 \div 3 = 63 \)
- 63 can also be divided by 3:
\( 63 \div 3 = 21 \)
- 21 can also be divided by 3:
\( 21 \div 3 = 7 \)
- 7 is a prime number.

So, the prime factorization of 567 is:
\( 3^4 \times 7^1 \)

#### Step 2.3: Prime factorization of 693
To find the prime factors of 693:
- 693 can be divided by 3:
\( 693 \div 3 = 231 \)
- 231 can also be divided by 3:
\( 231 \div 3 = 77 \)
- 77 can be divided by 7:
\( 77 \div 7 = 11 \)
- 11 is a prime number.

So, the prime factorization of 693 is:
\( 3^2 \times 7^1 \times 11^1 \)

### Step 3: Identify common factors
Now, we find the common prime factors from the factorizations:
- For 3: The minimum power is \( 3^2 \) (from 441 and 693).
- For 7: The minimum power is \( 7^1 \) (from 567 and 693).

### Step 4: Calculate the HCF
Now we multiply the common prime factors:
\[ HCF = 3^2 \times 7^1 = 9 \times 7 = 63 \]

### Conclusion
The greatest number that will divide 445, 572, and 699, leaving remainders 4, 5, and 6 respectively, is **63**.

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