Find the largest number by which dividing 310, 550 and 728 gives the same remainder.

Given:

Numbers: 310, 550, 728

Calculation:

Let the required number be N. When each number is divided by N, they leave the same remainder.

Step 1: Subtract the smaller numbers from the larger ones.

550 - 310 = 240

728 - 550 = 178

728 - 310 = 418

Step 2: Find the HCF of 240, 178, and 418.

Prime factorization:

240 = 2 × 2 × 2 × 2 × 3 × 5

178 = 2 × 89

418 = 2 × 11 × 19

Step 3: Identify the common factors.

The common factor is 2.

Step 4: The HCF is 2.

⇒ The greatest number is 2.

∴ The greatest number is 2