please I need some help on these qustion
1) Use the theory of congruences to show that [89|(2^44) − 1].
2) For any n 2 N find the remainder when the number
1^5 + 2^5 + 3^5 + · · · + (n − 1)^5 + n^5
is divided by 4.
Hello,
I don't know how far you've studied these things... Nevermind...
We want to show that
The cardinal of (Z/89Z)* is 88 because 89 is a prime integer.
The elements of (Z/89Z)* are the positive integers <89 and coprime with 89. Because 89 is prime, any integer <89 will be coprime with 89. Hence, 2 belongs to (Z/89Z)*.
We know that the order m of an element belonging to (Z/89Z)*, here 2, divides its cardinal, 88.
Hence,
m is such as :
is .
Now,
---->
QED.