Let the common divisor of given numbers and the remainder be m and n respectively. In modular arithmetic form, given number 670 is written as follows: 670≡r (mod m) ⇒ 670－r≡0 (mod m). In the same manner, 449－r≡0 (mod m), 364－r≡0 (mod m) and 313－r≡0 (mod m).
All the left sides of these expressions are divisible by m. Therefore, numbers obtained by the subtractions of any exp.s among them are also divisible by m: (670－r)－(449－r)≡(449－r)－(364－r)≡(364－r)－(313－r)≡0 (mod m) ⇒ 221≡85≡51≡0 (mod m).
By the subtraction rule, 85－51≡51－51 (mod m) ⇒ 51≡34≡0 (mod m). In the same manner, 51－34≡34－34 (mod m) ⇒ 34≡17≡0 (mod m) ⇒ 221≡85≡51≡34≡17≡0 (mod m). Here, 17 is a prime number that is not divisible by any number other than 1 and itself. Thus, the divisor is m=17 ⇒ 670÷17=39 r. 7, so the remainder is r=7.
CK: 449=17x26+7, 364=17x21+7 and 313=17x18+7 CKD. Therefore, the common divisor of given numbers is 17, and the remainder is 7.