# Thread: integer solutions of equation

1. ## integer solutions of equation

"Express all integer solutions in terms of n to the equation 840x+354y=6n, where n is natural." please
First I used the highest common factor method to find solutions for 840x+354y=6
solutions are
x=354m-8
y=19-840m
not sure what to do about 6n.

I really need an answer for this one please, anything you have that might help would be greatly appreciated.

2. gcd(840,354)=6
thus 6=a(840)+b(354) for some a,b both integers
this solves to a=-8 and b=19
therefore if you have 6n=840x+354y then i suppose x=-8n and y=19n

3. Originally Posted by jaco
if you have 6n=840x+354y then i suppose x=-8n and y=19n
That gives you one solution for each value of n. If you want all the solutions then you need to use the fact that the least common multiple of 354 and 840 is $\displaystyle 140\times354 = 59\times840\ (=49560)$. The general solution is then $\displaystyle x = 59k-8n$ and $\displaystyle y = 19n-140k$, for any integer k.