Find the greatest number that divides 125, 218, 280 and 342 so as to leave the…

Step-by-Step Solution

To find the greatest number that divides a set of numbers and leaves the same remainder in each case, follow this systematic approach:

1. Calculate the differences between the numbers: Given the numbers 125, 218, 280, and 342, find the differences between consecutive numbers or pairs:

   • 218 - 125 = 93

   • 280 - 218 = 62

   • 342 - 280 = 62

   • (You can also check pairs like 280 - 125 = 155, 342 - 218 = 124, etc., but consecutive differences are sufficient here).

2. Find the Highest Common Factor (HCF) of these differences: We now need the HCF of 93, 62, and 62.

   • Factors of 93: 1, 3, 31, 93

   • Factors of 62: 1, 2, 31, 62

   • The largest common factor shared by these differences is 31.

3. Conclusion: The greatest number that divides 125, 218, 280, and 342 to leave the same remainder is 31.