Question: "Find the least positive integer n that satisfies..." - Plainmath

Jason Farmer

Answered question

2021-01-31

Find the least positive integer n that satisfies the congruence

7^{128}

congruent to n (mod13)

Answer & Explanation

Luvottoq

Skilled2021-02-01Added 95 answers

Step 1
Given,

7^{128} \equiv n (mod 13)

Find the value of n that satisfies the above congruence.
Step 2

7^{128} \equiv n (mod 13)

Since 7 is not the multiple of 13.
Therefore,

7^{12} \equiv 1 (mod 13)

Then,

7^{128} \equiv n (mod 13)

\Rightarrow {(7^{12})}^{10} \times 7^{8} \equiv n (mod 13)

\Rightarrow {(1)}^{10} \times 7^{8} \equiv n (mod 13)

\Rightarrow 1 \times 7^{8} \equiv n (mod 13)

\Rightarrow 7^{8} \equiv n (mod 13)

Step 3
Also,

7^{2} \equiv 10 (mod 13)

So,

7^{8} \equiv n (mod 13)

\Rightarrow {(7^{2})}^{4} \equiv n (mod 13)

\Rightarrow {(10)}^{4} \equiv n (mod 13)

\Rightarrow {(10^{2})}^{2} \equiv n (mod 13) [∴ {(10)}^{2} \equiv 9 (mod 13)]

\Rightarrow {(9)}^{2} \equiv n (mod 13)

\Rightarrow 3 \equiv n (mod 13)

Step 4
Hence,
n=3

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