Introductory Problem For Number Congruence

image containing math problem statement
An introductory number congruence problem for beginners. Try solving this problem live and get a feel of the underlying theory.
We can solve this problem using the concepts of   .

$6^2 = 36 \equiv -1\ (mod\ 37)$

Therefore, $6^4 = (6^2)^2 \equiv (-1)^2\ (mod\ 37) \equiv 1\ (mod\ 37)$

Now, we can write $2023 = 2020 + 3 = 4 \times 505 + 3$

Therefore,
$6^{2023}\ (mod\ 37)$

$= \left[ 6^{4 \times 505 + 3} \right]\ (mod\ 37)$

$= \left[ (6^4)^{505} \cdot 6^3 \right]\ (mod\ 37)$

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$= \left[ (6^4)^{505} \cdot 6^3 \right]\ (mod\ 37)$

$= (6^4)^{505}\ (mod\ 37) \times 6^3 \ (mod\ 37)$

$= (1)^{505}\ (mod\ 37) \times (6^2 \times 6) \ (mod\ 37)$

$= 1 \times (-1) \times 6 \ (mod\ 37)$

$=  (-6) \ (mod\ 37)$

$= 31 \ (mod\ 37)$

Therefore, the remainder will be $31$