Find the largest number by which we can divide 63, 138, 228 so as to have same…

To find the largest number that divides 63, 138, and 228, leaving the same remainder in each case, we follow a specific procedure. Because we don't know what that remainder is, we eliminate it by calculating the differences between the numbers.

Step-by-Step Calculation

1. Calculate the differences between the numbers:

   • 138 - 63 = 75

   • 228 - 138 = 90

   • 228 - 63 = 165

2. Find the H.C.F. of these differences: We need the Highest Common Factor of 75, 90, and 165.

   • 75 = 3 × 5 × 5

   • 90 = 2 × 3 × 3 × 5

   • 165 = 3 × 5 × 11

   • The common factors present in all three are 3 and 5.

   • H.C.F. = 3 × 5 = 15.