Find the greatest number which will divide 478 and 719 leaving remainders 2 and 5 respectively

Options

A. 242
B. 240
C. 236
D. 238

Answer

To find the greatest number that will divide 478 and 719 leaving remainders of 2 and 5 respectively, we can follow these steps:

### Step 1: Adjust the Numbers
First, we need to adjust the two numbers by subtracting their respective remainders from them.

- For 478, subtract 2:
\( 478 - 2 = 476 \)

- For 719, subtract 5:
\( 719 - 5 = 714 \)

### Step 2: Find the HCF
Next, we need to find the highest common factor (HCF) of the adjusted numbers 476 and 714.

#### Step 2.1: Prime Factorization
To find the HCF, we can use the prime factorization method or the Euclidean algorithm. Here, we will use the Euclidean algorithm.

1. Divide the larger number by the smaller number and find the remainder:
\( 714 \div 476 = 1 \) (remainder 238)
\( 714 - 476 \times 1 = 238 \)

2. Now, take the smaller number (476) and divide it by the remainder (238):
\( 476 \div 238 = 2 \) (remainder 0)
\( 476 - 238 \times 2 = 0 \)

Since the remainder is now 0, the last non-zero remainder is the HCF.

### Step 3: Conclusion
The HCF of 476 and 714 is 238. Therefore, the greatest number that will divide 478 and 719 leaving remainders of 2 and 5 respectively is **238**.

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