Question 408264
As a corollary to Euclid's algorithm, there are integers s and t such that as + bt = gcd(a,b).  Hence if we're able to find integers s,t, such that 

s(5n + 3) + t(7n + 4) = 1, 

then we've shown that 5n + 3, 7n + 4 are relatively prime for all n.

==> (5s + 7t)n + (3s + 4t) = 1.

It's enough to see if the system 
5s + 7t = 0
3s + 4t = 1
has integer solutions.
Indeed, the solutions are s = 7, and t = -5.
Therefore 5n + 3, 7n + 4 are relatively prime for all n.