Find the greatest number that will divide 43,91,183 so as to leave the same remainder in each case

It is 4.

Step-by-step explanation:

If x is the greatest number and b the remainder the we have the equations:

ax + b = 183

cx + b = 91

dx + b = 43 where a, c and d are whole numbers.

Subtracting the last equation from the first:

ax - dx = 140

x (a - d) = 140

So x must be a factor of 140.

140 = 2 * 2 * 5 * 7

Trial and error:

Let x = 35:

43 / 35: remainder is 8

91 / 35 : rem = 21 So NOT 35.

x = 7: 43/7 rem = 1

91 / 7 rem = 0 So NOT 7.

x = 5: 43/5 rem = 3, 91/5 rem = 1, NOT 5.

x = 10 43/10 rem 3, 91/10 rem 1 NOT 10.

x = 20 43/20 rem 3, 91/20 rem 11 so NOT 20.

x = 28 43/28 rem 15, 91/28 rem 7 so NOT 28.

x = 14: 43 / 14 rem = 1, 91/14 rem = 7 NOT 14.

x = 4: 43/4 rem 3, 91/4 rem 3, 183 / 4 rem 3. So it is 4.