a number n divides each of 17 and 30 with the same remainder in each case. what is the largest value n can have

Question

a number n divides each of 17 and 30 with the same remainder in each case. what is the largest value n can have?

Answer 1

Using some simple number theory with modulus, let's make two equations:

 

30 = r + nA

17 = r + nB

 

Where r is the remainder and is the same in both equations, and A and B are both integers(Note: A and B can not be anything but whole numbers assuming n is a whole number). Now, let's multiply (-1) to both sides of 17 = r + nB and let's add both equations. As a result we will obtain: 13 = n(A - B) or (13/n) = A - B. Now let's say A -B = C where C is an integer or whole number since the difference of two whole numbers can only result to a smaller whole number. Thus, (13/n) = C, which means 13/n is a whole number as well and in order for that to happen then -13<= n <= 13. Since we're finding the largest value of n, this means n is 13 and furthermore the largest value that n can have when it divides both 17 and 30 and obtains the same remainder is 13.