Explanation:

To find the greatest number that divides 784 and 857, leaving remainders 4 and 2, we have to think about some basic principles.

- The number we seek must divide the difference between 784 and 4, which is 780.

- It also must divide the difference between 857 and 2, which is 855.

- Essentially, we need the greatest common divisor (GCD) of 780 and 855.

Let's consider your options:

- Option 1 (3): Check if 3 divides both 780 and 855 without remainder. It does, but is there a larger number?

- Option 2 (5): 5 divides both numbers, but it’s not the largest.

- Option 3 (30): 30 divides 780, but not 855.

- Option 4 (15): 15 divides both 780 and 855.

After checking, Option 4 (15) divides both numbers evenly.

Option 4 (15) is the greatest number that meets the conditions.

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