Using Fermat's theorem :
If p is a prime number and a, p are co primesĀ $$\left(a^{p-1}\right)\ mod\ p=1$$
Remainder whenĀ $$19^{20}$$ is divided by 7 =Ā $$19^2$$ mod 7 =4. ( HereĀ $$19^{20\ }=\ \left(\left(19\right)^6\right)^3\cdot\left(19\right)^2$$
Since the remainder for $$19^6$$ is 1 the remainder forĀ $$\ 19^{20}$$ isĀ equivalent to theĀ $$\frac{19^2}{7}$$ = 4.
Remainder when $$20^{19}$$ is divided by 7 = $$20^1$$ mod 7 =6.( HereĀ $$\frac{20^6}{7}$$ the remainder is 1 and sinceĀ
$$20^{19}=\ \left(20^6\right)^3\cdot\left(20\right)^1\ =\ \frac{\left(1\cdot20\right)}{7}$$. The remainder is 6.
Remainder when $$19^{20} - 20^{19}$$ is divided by 7=4-6=-2=> 5.Ā
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