Question

calculate 5²⁰³⁹ mod 41
urgent​

Answer 1

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

Answer 2

The 'Reduce', 'Reuse' and 'Recycle' elements are referred to as the '3 R's' Reducing - Try to reduce the amount of waste you produce, as this is the best way to help the environment!