calculate 5²⁰³⁹ mod 41
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Answer:
Since 41 is a prime, and 5 is prime to 41, therefore by Fermat's theorem
5
40
=1(mod41)
By division algorithm,
2039=50.40+39
Therefore,
5
2039
=5
50.40+39
=(5
40
)
50
.5
39
≡1
50
.5
39
(mod41)
≡5
39
(mod41)
To calculate 5
39
(mod41), we first calculate 5
39
(mod41) where n=2,4,8,16,32
5
2
≡25≡−16(mod41)
5
4
≡256≡10(mod41)
5
8
≡100≡18(mod41)
5
16
≡324≡−4(mod41)
5
32
≡16(mod41)
Now, 5
39
≡5
32
.5
4
.5
2
.5=16.10.(−16).5
≡33(mod41)
Thus 5
2039
(mod41)=33
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