The greatest number that will divide 398, 436 and 542 leaving 7, 11 and 15 as remainders, respectively, is :

Options

A. 16
B. 17
C. 18
D. 19

Answer

To find the greatest number that will divide 398, 436, and 542 leaving remainders of 7, 11, and 15 respectively, we can follow these steps:

### Step 1: Subtract the remainders from each number
We need to subtract the remainders from each of the given numbers to find the numbers that will be divisible by the greatest number we are looking for.

- For 398 with a remainder of 7:
\[ 398 - 7 = 391 \]

- For 436 with a remainder of 11:
\[ 436 - 11 = 425 \]

- For 542 with a remainder of 15:
\[ 542 - 15 = 527 \]

### Step 2: Find the differences between the modified numbers
Next, we will find the differences between the modified numbers we obtained in Step 1.

- Difference between 425 and 391:
\[ 425 - 391 = 34 \]

- Difference between 527 and 425:
\[ 527 - 425 = 102 \]

- Difference between 527 and 391:
\[ 527 - 391 = 136 \]

### Step 3: Find the greatest common divisor (GCD)
Now, we need to find the GCD of the differences calculated in Step 2 (34, 102, and 136).

1. Find the GCD of 34 and 102:
- The factors of 34: 1, 2, 17, 34
- The factors of 102: 1, 2, 3, 6, 17, 34, 51, 102
- Common factors: 1, 2, 17, 34
- GCD(34, 102) = 34

2. Now find the GCD of 34 and 136:
- The factors of 136: 1, 2, 4, 8, 17, 34, 68, 136
- Common factors with 34: 1, 2, 17, 34
- GCD(34, 136) = 34

Thus, the GCD of 34, 102, and 136 is 34.

### Step 4: Conclusion
The greatest number that will divide 398, 436, and 542 leaving remainders of 7, 11, and 15 respectively is:
\[ \boxed{34} \]