User:Jcgonzal/Sandbox

We have discussed four ways of "saying the same thing", related to congruences. For example, the following four sentences are equivalent:

1. 20 - 11 is a multiple of 3. 2. If you divide 20 by 3, and you divide 11 by 3, you get the same remainder. 3. 20 \equiv 11, mod 3. 4. \overline{20} = \overline{11} (modulo 3).

Each of the following five sentences is written in one of the four forms above. Translate each of the following four sentences into the other three forms.

* 17 - (-3) is a multiple of 5. * If you divide x^2 by 6, and you divide y by 6, you get the same remainder. * m^2 \equiv 0, mod m.   * \overline{3x + 2} = \overline{5} (modulo 7).

Computation Exercises, 3 Points

In each of the following problems, carry out a computation in modular arithmetic. Your answer should have the form \bar a, where a is a natural number between 0 and the modulus. After carrying out the computation, interpret your result using a sentence involving the words "multiple" or "divide" and "remainder". The following is an example:

* \overline{11}^{12}, modulo 13.

Computation: \overline{11}^{12} = \overline{-2}^{12} = ((\overline{-2})^4)^3 = \overline{16}^3 = \overline{3}^3 = \overline{27}=\overline{1}. Interpretation: If you divide 11^{12} by 13, you get a remainder of 1. Alternatively, 11^{12} - 1 is a multiple of 13.

Here are the problems:

* \overline{11} \cdot \overline{13}, modulo 15. * \overline{9}^{1020}, modulo 40. * \overline{3}^{10} - \overline{2}^{10}, modulo 11.

Solving a Linear Congruence, 3 Points

Find an integer x, between 0 and 46, such that:

* \overline{36} \overline{x} + \overline{11} = \overline{35}, modulo 47.

Show your work, including the Euclidean algorithm. No Solutions, 2 Points

Prove that there are no solutions to the Diophantine equation x^6 + y^6 = 700003. Use the following strategy:

* What are the possible remainders if you divide x^6 or y^6 by 7? o What is \bar 1^6? What is \bar 2^6? etc.. * What are the possible remainders if you divide x^6 + y^6 by 7? * What is the remainder if you divide 700003 by 7?