Solving a Diophantine equation?
63* #63*x+70*y+75*z=91#
63*
2 Answers
for any integers
Explanation:
Given:
#63x+70y+75z=91#
First note that all terms are divisible by
So we require
Then:
#9x+10y+75w=13#
Note that
Hence we require
Hence
Then:
#45k+18+10y+75w=13#
So:
#9k+2y+15w+1=0#
Since
Then:
#0 = 9k+2y+15(k+2h+1)+1 = 24k+2y+30h+16#
So:
#12k+y+15h+8 = 0#
Then note that
Then:
#4k+f+5h = 0#
So for any integers
Then:
#y = 3f-8 = 3(-4k-5h-8) = -12k-15h-8#
and we have a solution of the original equation, with:
#{ (x = 5k+2), (y = -12k-15h-8), (z = 7k+14h+7) :}#
See below.
Explanation:
This equation can be solved using a technique inspired on the sieve of Eratosthenes.
1)
now calling
2)
now calling
3)
and now calling
4)
but
now
Resuming