<s>[INST] Computing greatest common divisor and determining whether Diophantine equation has solutions Question:
(1) Compute gcd$(224, 12 + 2^{10})$
(2) Describe in words the set of all integers $ n$ for which the equation $ \ 4x + ny = 1$ has integer solutions. 
(3) 
Describe in words the set of all integers $ \ n$ for which the equation $ \ 3x + ny = 1$ has integer solutions. 
My attempt:
For (1) the Euclidean algorithm will be difficult to use. Should I use the theorem that states gcd$ \ (a,b) = $ gcd($ a -bq, b)$ or should I prime factorize both sides? I am not quite sure.
For (2) we need need gcd$(4,n) = 1$ in order for the equation to have integer solutions. This means that $ \ n$ must belong to the set of all odd integers. 
For (3) we need need gcd$(3,n) = 1$ in order for the equation to have integer solutions. This means that $ \ n$ must belong to the set of all integers that are not divisible by 3. 
 [/INST] $224= 4\cdot 7\cdot 8$ and $12+2^{10} = 4(3+2^8)=4\cdot 7\cdot 37$. 
So $gcd(224,12+2^{10}) =28$.
The rest is fine.
Ok, ok, we can factor $$2^8+3 = 2^8-2^2+4+3= 2^2(2^6-1)+7 = 4(2^3-1)(2^3+1)+7 =4\cdot 7\cdot 9+7 = 7(36+1) =7\cdot 37$$
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