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 to leave the same remainder in each case, you must find the Highest Common Factor (H.C.F.) of the differences between these numbers.

Step-by-Step Calculation
1. Calculate the differences between the numbers:
To eliminate the unknown common remainder, we find the differences between the pairs:

Difference 1: 138 - 63 = 75

Difference 2: 228 - 138 = 90

Difference 3: 228 - 63 = 165

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

Prime factorization:

75 = 3 * 5 * 5 = 3 * 5^2

90 = 2 * 3^2 * 5

165 = 3 * 5 * 11

The H.C.F. is found by taking the lowest power of the common prime factors:

Common factors: 3^1 and 5^1

H.C.F. = 3 * 5 = 15.