Suppose that you are given positive integers a, b, c. Prove that if there are any integers
x, y satisfying ax + by = c then there are infinitely many solutions to this equation.
Attempt:(edited)
If a and b are coprime you can always find integers p, q such that

ap + bq = 1 using Euclid's algorithm.

Then multiply by c to get

ax' + by' = c ......(1) where x' = cp, y' = cq

and finally let x = x'+bt y = (y'-at) and substituting into the
equation

ax + by = c we get

a(x'+bt) +b(y'-at) = c

Suppose $x$ is a binary string of length $n - k$, where $n, k$ are integers and $n > k$. An 'extension' of $x$ is a string of length $n$ which is formed from $x$ by inserting exactly $k$ total new 0s and 1s at any position in the string $x$. How many distinct such 'extensions' are there from $x$? Does it depend on more than the length of the string $x$ (i.e., the value of $k$)?
Based on numerical evidence the answer is 'it does not depend on the actual string x'.And the number of distinct extensions is $\sum_0^k \binom{n}{i}$