What is the greatest number that divides 13850 and 17030 leaves a remainder 17?

Options

A. 477
B. 159
C. 107
D. 87

Answer

To find the greatest number that divides both 13850 and 17030 leaving a remainder of 17, we can follow these steps:

### Step 1: Adjust the Numbers
Since we want the greatest number that leaves a remainder of 17, we first subtract 17 from both numbers.

- For 13850:
\[ 13850 - 17 = 13833 \]

- For 17030:
\[ 17030 - 17 = 17013 \]

### Step 2: Find the HCF
Now, we need to find the Highest Common Factor (HCF) of the adjusted numbers 13833 and 17013.

### Step 3: Apply the Euclidean Algorithm
We will use the Euclidean algorithm to find the HCF. The process involves repeated division:

1. Divide the larger number by the smaller number:
\[ 17013 \div 13833 \]
The quotient is 1, and we find the remainder:
\[ 17013 - 13833 \times 1 = 17013 - 13833 = 3170 \]

2. Now, we take the smaller number (13833) and divide it by the remainder (3170):
\[ 13833 \div 3170 \]
The quotient is 4, and we find the remainder:
\[ 13833 - 3170 \times 4 = 13833 - 12680 = 1153 \]

3. Next, we take 3170 and divide it by 1153:
\[ 3170 \div 1153 \]
The quotient is 2, and we find the remainder:
\[ 3170 - 1153 \times 2 = 3170 - 2306 = 864 \]

4. Now, we take 1153 and divide it by 864:
\[ 1153 \div 864 \]
The quotient is 1, and we find the remainder:
\[ 1153 - 864 \times 1 = 1153 - 864 = 289 \]

5. Next, we take 864 and divide it by 289:
\[ 864 \div 289 \]
The quotient is 2, and we find the remainder:
\[ 864 - 289 \times 2 = 864 - 578 = 296 \]

6. Now, we take 289 and divide it by 296:
\[ 289 \div 296 \]
The quotient is 0, and we find the remainder:
\[ 289 - 296 \times 0 = 289 \]

7. Finally, we take 296 and divide it by 289:
\[ 296 \div 289 \]
The quotient is 1, and we find the remainder:
\[ 296 - 289 \times 1 = 296 - 289 = 7 \]

8. Now, we take 289 and divide it by 7:
\[ 289 \div 7 \]
The quotient is 41, and we find the remainder:
\[ 289 - 7 \times 41 = 289 - 287 = 2 \]

9. Finally, we take 7 and divide it by 2:
\[ 7 \div 2 \]
The quotient is 3, and we find the remainder:
\[ 7 - 2 \times 3 = 7 - 6 = 1 \]

10. Lastly, we take 2 and divide it by 1:
\[ 2 \div 1 \]
The quotient is 2, and the remainder is 0.

### Step 4: Conclusion
When we reach a remainder of 0, the last non-zero remainder is the HCF. In this case, the last non-zero remainder is 1.

### Final Answer
Thus, the greatest number that divides both 13850 and 17030 leaving a remainder of 17 is:
\[ \text{Greatest number} = 159 \]