(Short question) Modular inverse, there is no integer solution

Question

i just want to convince my self.

Does $10^{-1}\pmod {26}$ not have an integer solution?

Answer 1

I guess what you mean is to find an integer x such that $10x\equiv 1(mod 26)$
This is the same as finding $x,y\in \mathbb{Z}$ such that $10x+26y=1$
However, since $\gcd(10,26)=2$, the least positive element in the set $\{10x+26y: x,y\in \mathbb{Z}\}$ is $2$.
Thus there is no integral solution.