Modular arithmetic

Question

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324

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What is the smallest positive integer n such that 3n = 1356 (mod 27).

Guest Jan 26, 2022

Guest · Answer 1 · 2022-01-27T03:56:10

This is equivalent to n=452 mod 27 so the answer is 452

Melody · Answer 2 · 2022-01-27T05:33:48

What is the smallest positive integer n such that 3n = 1356 (mod 27).

$3n=27k+1356\\ 3n=3*9k+3*452\\ n=9k+452\\~\\ 9k+452>0\\ 9k>-452\\ k>-50.2\\~\\ when\;k=-50\\ n=9*-50+452\\ n=2$

the smallest value of n is 2

check

3*2=6 (mod 27)

1356 = 50+6/27 = 6 mod(27)

LaTex:

3n=27k+1356\\
3n=3*9k+3*452\\
n=9k+452\\~\\
9k+452>0\\
9k>-452\\
k>-50.2\\~\\
when\;k=-50\\
n=9*-50+452\\
n=2