Question 1162454
What is the greatest integer, which when divided into 383, 527, or 815
leaves the same remainder?
<pre>
Let the greatest integer be d, and the common remainder be r.  Then
Let q<sub>1</sub>, q<sub>2</sub>, q<sub>3</sub> be the quotients. Then

{{{system(q[1]d+r=383,q[2]d+r=527, q[3]d+r=815)}}}

Solving the first two equations for r:

{{{d=(383-527)/(q[1]-q[2])=(-144)/(q[1]-q[2])=144/(q[2]-q[1])}}}

That tells us that the most d can be is 144, and it can be 144 if we can get
the denominator q<sub>2</sub>-q<sub>1</sub> to equal 1.

Solving the first and third equations for r:

{{{d=(383-815)/(q[1]-q[3])=(-432)/(q[1]-q[3])=432/(q[3]-q[1])=(3*144)/(q[3]-q[1])}}}

Solving the second and third equations for r:

{{{d=(527-815)/(q[2]-q[3])=(-288)/(q[2]-q[3])=288/(q[3]-q[2])=(2*144)/(q[3]-q[2])}}}

So d can be 144 if we can have 

q<sub>2</sub>-q<sub>1</sub> = 1
q<sub>3</sub>-q<sub>1</sub> = 3
q<sub>3</sub>-q<sub>2</sub> = 2

Solving that dependent system, we get

{{{(matrix(1,5,q[1],",",q[2],",",q[3]))}}}{{{""=""}}}{{{(matrix(1,5,q[3]-3,",",q[3]-2,",",q[3]))}}}

So if we let q<sub>3</sub>=4, we have

{{{(matrix(1,5,q[1],",",q[2],",",q[3]))}}}{{{""=""}}}{{{(matrix(1,5,1,",",2,",",4))}}}

We can have the maximum of {{{d=144/(2-1)=144}}}

Edwin</pre>