Question #24690

1 Answer
Nov 22, 2016

Answer:

Considering only positive solutions,

#x=7,y=32#
#x=18,y=17# and
#x=29, y = 2#

Explanation:

#30x+22y=914# has the same solutions of

#15x+11y=457# because #2(15x+11y)=2 cdot457#

Now, by Bézout's identity,
https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity

#ax+by=c# has integer solution if and only if #d = gcd(a,b)# is a divisor of #c#

In our case, #d = 1# and #c = 457# is prime, but #1# is a universal divisor so the equation must have solutions.

#x=7,y=32#
#x=18,y=17# and
#x=29, y = 2#

Considering only positive solutions.