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LCM equation (Posted on 2013-11-19) Difficulty: 2 of 5
The least common multiple (LCM) of 2 numbers is the smallest number that they both divide evenly into.
e.g.:
LCM(8,10)=40
LCM(17,11)=187

For any integer n, show that LCM(9n + 8, 6n + 5) = 54n^2 + 93n + 40.

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts re: Proof | Comment 3 of 5 |
(In reply to Proof by Charlie)

Charlie:


I don't think that your hand waving actually constitutes an accurate proof.  While I don't follow it exactly, it seems that I could follow the same steps to "prove" that (12n - 2) and (8n + 3) are relatively prime, but this not in fact the case for all n.  When n = 11, for instance, both are divisible by 13.

Using my method, it is seen that any factor of (12n - 2) and (8n + 3) is also a factor of (12n - 2) - (8n - 3) = (4n  - 5).  But any factor of these three is also a factor of (8n - 3) - (4n - 5) = (4n + 8).  But any factor of these 4 is also a factor of (4n + 8) - (4n - 5) = 13.  Therefore, for all n, the GCD of (12n - 2) and (8n - 3) is either 1 or 13.



   

Edited on November 20, 2013, 1:28 am
  Posted by Steve Herman on 2013-11-20 01:27:19

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