Find the least positive integer n that satisfies the congruence 7^128 congruent to n (mod13)

Jason Farmer 2021-01-31 Answered
Find the least positive integer n that satisfies the congruence
7128 congruent to n (mod13)
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Expert Answer

Luvottoq
Answered 2021-02-01 Author has 95 answers
Step 1
Given,
7128n(mod13)
Find the value of n that satisfies the above congruence.
Step 2
7128n(mod13)
Since 7 is not the multiple of 13.
Therefore,
7121(mod13)
Then,
7128n(mod13)
(712)10×78n(mod13)
(1)10×78n(mod13)
1×78n(mod13)
78n(mod13)
Step 3
Also,
7210(mod13)
So,
78n(mod13)
(72)4n(mod13)
(10)4n(mod13)
(102)2n(mod13)[(10)29(mod13)]
(9)2n(mod13)
3n(mod13)
Step 4
Hence,
n=3
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