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

Key idea: The greatest number that divides all given numbers leaving the same remainder equals the greatest common divisor (GCD) of the differences between those numbers.

Step 1: Compute the differences between the numbers.

• 218 − 125 = 93

• 280 − 218 = 62

• 342 − 280 = 62

• 280 − 125 = 155

• 342 − 125 = 217

• 342 − 218 = 124

Step 2: Take the GCD of these differences.

1. gcd(93, 62): 93 = 62 × 1 + 31, so gcd(93, 62) = gcd(62, 31) = 31.

2. Check 31 against the remaining differences: 155 ÷ 31 = 5, 217 ÷ 31 = 7, 124 ÷ 31 = 4. So 31 divides all differences.

Conclusion: The greatest number that divides 125, 218, 280 and 342 leaving the same remainder in each case is 31.