So i just got a hang of congruence but I am unable to solve this also how does one find all integers solution for $1566x - 232y = 116$ such that $0 \leq x,y \leq 100$
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It is a particular case of a very classical issue: see for example (http://math.stackexchange.com/q/20717) – Jean Marie Nov 02 '16 at 07:50
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$$ (3996,3071)=37$$ which does not divide $$482$$
$$108x-83y=\dfrac{482}{37}$$
The left hand side is an integer unlike the right hand side.
As $(1566,232,116)=58$
$$1566x-232y=116\iff27x-4y=2 \iff 27(x-2)=4(y-13)\iff y-13=\dfrac{27(x-2)}4$$
As $(27,4)=1,4|(x-2)\iff x=4m+2$ where $m$ is any integer
Consequently, $y-13=\dfrac{27(x-2)}4=?$
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Thanks! can you explain me $289(4^{2016})-1000$ is divisible by 68? How does one do without calculator? – Arya Nov 02 '16 at 07:05
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