Fermat's Little Theorem Variation Problem

Question

I've seen this type of problem circulated around on this high school math competition called Number Sense, where the aim is to solve as many problems as fast as possible and more importantly, all mentally. The type of problem looks something like the following: $$20^6\div11\mathrm{\space has\space a \space remainder\space of\space ?}$$ My first approach was to this problem was saying that we can utilize the fact that $20^{10}\div 11\equiv 1\pmod{11}$ and essentially say that now if we take the square root of this, we get $20^5\div 11\equiv 1\pmod{11}$, and therefore $20^6\pmod{11}=20\pmod{11}=9$, giving us 9 as the answer. However, I have noticed that this fails to work in some instances. Take the following problem: $$8^7\div 13\mathrm{\space has\space a\space remainder\space of\space ?}$$ In this instance, if we take the exact same approach, utilizing $8^{12}\div 13\equiv 1\pmod{13}$, when we take the square root of $8^{12}$ we get $8^6$ so theoretically, $8^6\div 13\equiv 1\pmod{13}$ which can then lead us to $8^6\times 8\div 13$ having a remainder of 8. I noticed that whenever you plug something like this into Wolfram Alpha though, that the answer seems to be 5, and the reason being for this is that $8^6\div 13\equiv -1\pmod{13}$ rather than positive 1 which would make the remainder the 5 that we see in Wolfram Alpha rather than the 8 that we expect. Can anybody think of a method to approach this problem in terms of how we are able to solve for if the modulus will be a negative 1 or a positive 1? I have been trying to figure this out for a day or two now but being a high-school student that doesn't know a whole lot about things like this, I haven't been able to come to some solid conclusions (though I have had ideas). Other posts, such as the one referenced with the $2^11\div 23$ question have had some responses but have only clarified how you can get to the fact that the mod is -1 or +1, rather than how you can actually determine which one it is to get the correct answer to the problem. I'm not sure if determining whether it is -1 or +1 lies in Fermat's Little Theorem, but all ideas are welcome. I've also looked into Euler's Toitent function but that has proved to be little help in solving the problem as well. Thanks to all!