Find the greatest number that will divide 43, 91 and 183 so as to leave the…

Key idea: If a number divides 43, 91 and 183 leaving the same remainder, then that number must divide the differences between the numbers.

• Compute the pairwise differences: 91 - 43 = 48, 183 - 91 = 92, and 183 - 43 = 140.

• The required greatest number is the greatest common divisor (HCF) of 48, 92 and 140.

• Compute gcd(48, 92) using the Euclidean algorithm: 92 = 48·1 + 44, 48 = 44·1 + 4, 44 = 4·11 + 0, so gcd(48, 92) = 4.

• Now gcd(4, 140) = 4, so gcd(48, 92, 140) = 4.

Therefore the greatest number that will divide 43, 91 and 183 leaving the same remainder is 4.