Phone Codes

The greatest number which can divide $x, y,$ and $z$ and leave the same remainder in each can be given as the greatest common factor of $\left|x - y\right|, \left|y - z\right|,$ and $\left|z - x\right|$ . so the number in this case would be greatest common factor of $(4387 - 2153), (7738 - 4387),$ and $(7738-2153)$ , which is the greatest common factor of 2234, 3351, and 5585. This greatest common factor is 1117.

let n be the answer (7738-4387) - (4387-2153) = n , 3351 - 2235 = n , 1117=n

rewrite: 7738-4387=ab & 4387-2153=ac solve integer a=1117 therefore fourth number is 1117

All give a remainder of 1036 when divided by 1117.

By Euclid's division algorithm, we know that a= bq+r, where a,b are positive integers and q,r are whole numbers and r=0 or is less than q

Thus,
2153= bq+r (1)
4387= cq+r (2)
7738=dq+r (3)
(2)-(1)= 2234=q(c-b)
Thus, q=(2234)/(c-b)
We observe that for 2234 to be a natural 4 digit number c-b=2 or 1 Clearly 1 is not possible, since q is less than or equal to 2153 Thus, q= 2234/2=1117 ans.