If R is the remainder when each of the numbers 1059, 1417, and 2312 is divided by D, where D is an integer greater than 1, compute the value of D - R.

Answer 1

Let p,q,r,s,t be some arbitrary integers, then

A=1059 = pD+R
B=1417 = qD+R
C=2312 = rD+R

E = B-A = sD = 358 = 2*179
F = C-B = tD = 895 = 5*179

Using Euclid's algorithm, all common factors between E and F are also common factors of (A-R),(B-R) and (C-R).

Thus 179 is the (only) common factor between the three numbers (A-R), (B-R) and (C-R), from which we determine that D=179 and R=164.
D-R=179-164=15.

Answer 2

1417-1059=358=2*179
2312-1417=895=5*179
D=179
R=164
D-R=15