SOLUTION: What is the greatest integer, which when divided into 383, 527, or 815 leaves the same remainder?

Let N be that number.


Then 383 divided by N, 527 divided by N and 815 divided by N give the same remainder.



It implies that the difference  527 - 383 = 144  is divisible by N

            and the difference  815 - 527 = 288  is divisible by N.



So, presumably N = 144, and we must check it.



    383 divided by 144 gives the remainder 95.

    527 divided by 144 gives the remainder 95.

    815 divided by 144 gives the remainder 95.



ANSWER.  The number under the question is 144.

Let the greatest integer be d, and the common remainder be r.  Then
Let q1, q2, q3 be the quotients. Then



Solving the first two equations for r:



That tells us that the most d can be is 144, and it can be 144 if we can get
the denominator q2-q1 to equal 1.

Solving the first and third equations for r:



Solving the second and third equations for r:



So d can be 144 if we can have 

q2-q1 = 1
q3-q1 = 3
q3-q2 = 2

Solving that dependent system, we get



So if we let q3=4, we have



We can have the maximum of 

Edwin