9.5k views
1 vote
True or False? Justify your answer. There exists r∈Z such that 5636¹⁷≡1 mod 9

User Odell
by
8.3k points

1 Answer

3 votes

False.

To justify the answer, it is necessary to solve the congruence relation 5636^17 ≡1 mod 9.

First, it's recognized that for any positive integer n, n^1 is congruent to n mod 9. This step is fairly straightforward, as it simply states that for all positive integers n, the number n will give the same remainder as n when divided by 9. It also has to be observed that any number can be expressed in mod 9 using its digit sum due to mathematical property of 9.

Taking 5636 mod 9, we find that the remainder r when 5636 is divided by 9 is equivalent to summing its digits and taking mod 9. This value of r is the one that will be used for further calculations.

Then, it's necessary to check whether r to the power of 17 is congruent to 1 mod 9. This is based on the fact that if 5636^17 is congruent to 1 mod 9, then r^17 should also be congruent to 1 mod 9.

So, the congruence r^17 is computed and then checked if it's equivalent to 1 modulo 9.

If the congruence equation holds true, then there exists an integer r such that the equation 5636^17 ≡1 mod 9 holds true.

However, after performing these calculations, we find that r^17 is not congruent to 1 mod 9. Therefore, the statement that there exists an integer r such that 5636^17 ≡1 mod 9 is false.

User KorbenDallas
by
8.2k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories