Question

When 19779 and 17997 are divided by a certain number they leave the same reminder. Find the largest such divisor and reminder. How many such divisor are there?

Answer 1

Divisor = d, remainder = r
19779 = kd + r
17997 = md + r

or, 1782 = (k-m)d

factors of 1782 are 1, 2, 3,6, 9, 11,18,22,27,33,54,66,81,99,162,198,297,594,891,1782
So, the largest such divisor will be 1782 and remainder will be 177 (Divide both 19779 and 17997, we get the remainder 177)
Among these factors, 1,3 completely divide 17997 and 19779 leaving reamder zero. So, such divisors are 2,6, 9, 11,18,22,27,33,54,66,81,99,162,198,297,594,891,1782 that is there are 18 such divisors which leave common remainder. Among them, 5 divisors 198,297,594,891,1782 will leave common remainder 177.