Find all integers x,y such that 2x+3y=185 and xy>x+y.
Problem
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Tags: number theory
Richie
08.04.2021 08:32
I may be wrong but I am getting 29 solutions of the equation.
Wildabandon
08.04.2021 09:23
jasperE3 wrote: Find all integers x,y such that 2x+3y=185 and xy>x+y.
From 2x+3y=185, we have (x,y)=(−185+3k,185+2k) where k∈Z. Because xy>x+y, we have 6k2+180k−1852>0. Then,
k>5√8538−906ork<−5√8538−906.Because k∈Z, =we conclude that k≥62 or k≤−93. So, the solutions are
(x,y)=(−185+3k,185+2k),k∈Z,k∈(−∞,−93]∪[62,∞).
cosmicc
23.05.2021 12:13
Wildabandon wrote: jasperE3 wrote: Find all integers x,y such that 2x+3y=185 and xy>x+y.
From 2x+3y=185, we have (x,y)=(−185+3k,185+2k) where k∈Z. Because xy>x+y, we have 6k2+180k−1852>0. Then,
k>5√8538−906ork<−5√8538−906.Because k∈Z, =we conclude that k≥62 or k≤−93. So, the solutions are
(x,y)=(−185+3k,185+2k),k∈Z,k∈(−∞,−93]∪[62,∞).
62 for k doesn't work
Pal702004
24.05.2021 18:06
2x+3y=185
Integer solutions:
y=1+2k,x=91−3k,k∈Z
xy>x+y⇔(x−1)(y−1)>1
(90−3k)(2k)>1⇒(30−k)k>0
1≤k≤29