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 April 10th, 2018, 08:27 AM #1 Newbie   Joined: Apr 2018 From: Algeria Posts: 3 Thanks: 0 ax+by=c equation, and numeric system problem. We have the equation: (E):5x-6y=3 1-Solve in Z^2 the equation (I already did, the solution is (x,y)=(6k+3 , 5k+2) (k∈Z)) 2- a and b are natural numbers: a=1α0α00 in the 3-base system. b=αβ0α in the 5-base system. -Find α and β , knowing that (a,b) is a solution to (E) Help please with the last question. April 10th, 2018, 09:29 AM #2 Global Moderator   Joined: Dec 2006 Posts: 21,029 Thanks: 2259 (α, β) = (2, 4) April 10th, 2018, 10:16 AM   #3
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Quote:
 Originally Posted by skipjack (α, β) = (2, 4)
But how? April 10th, 2018, 10:41 AM #4 Global Moderator   Joined: Dec 2006 Posts: 21,029 Thanks: 2259 Trial and error. There are only three possibilities for the value of α, of which only α = 2 leads to a solution. April 10th, 2018, 10:46 AM   #5
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Quote:
 Originally Posted by skipjack Trial and error. There are only three possibilities for the value of α.
Actually it's just 2, but I'm looking for a mathematical way to solve, thanks anyway. April 10th, 2018, 02:53 PM #6 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 673 Thanks: 88 What does solving in Z^2 mean? April 10th, 2018, 04:31 PM   #7
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Quote:
 Originally Posted by EvanJ What does solving in Z^2 mean?
I'll give you the (very) simple run-down. $\displaystyle \mathbb{Z} ^2$ is the set of all ordered pairs (a, b) where a and b are both integers.

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