All About Algebra
m 1 − n 1 = 6 1 . How many integral ordered pairs ( m , n ) are there?
The answer is 17.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
2 solutions
Moderator note:
Simon's favorite factorizing trick is the best approach to solving such Diophantine equations.
(Taken from Eamon's comment from a deleted solution)
The entire list is ( 5 , 3 0 ) ( 4 , 1 2 ) ( 3 , 6 ) ( 2 , 3 ) ( − 3 , − 2 ) ( − 6 , − 3 ) ( − 1 2 , − 4 ) ( − 3 0 , − 5 ) ( 7 , 4 2 ) ( 8 , − 2 4 ) ( 9 , − 1 8 ) ( 1 0 , − 1 5 ) ( 1 2 , − 1 2 ) ( 1 5 , − 1 0 ) ( 1 8 , − 9 ) ( 2 4 , − 8 ) ( 4 2 , − 7 )
It should be written that how many integral ordered pairs are there. If you don't mention that then the answer is infinite.
I believe the answer should be 9 and I am going to mention the possible solutions of this problem, let me know if I missed anything ( − 6 , − 3 ) ( 3 , 6 ) ( 7 , − 4 2 ) ( 8 , − 2 4 ) ( 9 , − 1 8 ) ( 1 2 , − 1 2 ) ( 1 8 , − 9 ) ( 2 4 , − 8 ) ( 4 2 , − 7 )
Log in to reply
This is my entire list: ( 5 , 3 0 ) ( 4 , 1 2 ) ( 3 , 6 ) ( 2 , 3 ) ( − 3 , − 2 ) ( − 6 , − 3 ) ( − 1 2 , − 4 ) ( − 3 0 , − 5 ) ( 7 , 4 2 ) ( 8 , − 2 4 ) ( 9 , − 1 8 ) ( 1 0 , − 1 5 ) ( 1 2 , − 1 2 ) ( 1 5 , − 1 0 ) ( 1 8 , − 9 ) ( 2 4 , − 8 ) ( 4 2 , − 7 )
In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the “dot dot dot” menu in the lower right corner. This will notify the problem creator who can fix the issues.
The equation can be written as 6 n − 6 m = m n
m n − 6 n + 6 m − 3 6 = − 3 6
( m − 6 ) ( n + 6 ) = − 3 6
Now 3 6 = 2 2 × 3 2 so it has ( 2 + 1 ) ( 2 + 1 ) = 9 factors, however, since − 3 6 must have at least 1 negative factor, then it has 9 × 2 = 1 8 factors (including negative factors).
With values m = 0 and n = 0 :
( 0 − 6 ) ( 0 + 6 ) = − 3 6
But m , n = 0 since it would make the original expression undefined.
So in total there are 1 8 − 1 = 1 7 integral solutions for ordered pairs ( m , n ) .