Solve a Congruence of a Number with Big Powers using Eulers Theorem

This is a tutorial on Number Theory in which we use Eulers theorem to solve a congruence of a number with extraordinary big powers. A number such as 7^7^37 will have far too many digits to realistically divide by 19.
So, we use properties of the Totient function or the phi function as it is sometimes known
Ceiling [7^37 Log [10, 7]]
15686807783369383792904017547009 decimal digits ≈ 1.56868×10^31 decimal digits
last 5 digits 83543
#algebra
#euclidsalgorithm
#bigpower
#mathematics
#eucliddivisionalgorithem
#euclidean
#mathtricks
#eulerstheorem
#numbersystem
#numbers

[MyOneFiftiethOfADollar]

Fairly certain your result is a special case of the identity 7^(7^x) == 7 (mod 19). for all natural numbers x. Fermat's little theorem gets the job done in lieu of Euler's Totient. Will double check later on.

[omarmenjivar]

Still does not make sense... This problem always looks like someone is just choosing random numbers one after the other.