I am having some repeated trouble getting the correct answer on linear congruences. Consider the following
12x \equiv 1 \pmod {77}
12 and 77 are relatively prime so this congruence has a solution. We search for a linear combination of 12 and 77 using the extended Euclidean algorithm.
77=12(6)+5\\ 12=5(2)+2\\ 5=2(2)+1
We now solve for the remainders
1=5-2(2)\\ 2=12-5(2)\\ 5=77-12(6)
Back substituting we find
1=77(5)+12(-32)
The solution to this congruence is 45 and -32 \equiv 45 (mod 77). What am I failing to do properly as to get the first positive solution?
Answer
Your answer is completely correct. More generally, the solution is 45+77k: k\in \mathbb{Z} as stated above, because
45+77k\equiv 45\mod 77.
If you're concerned about getting the negative answer first, it is simple to just add 77 as you did to find the first positive value.
No comments:
Post a Comment