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Introduction to the Theory of Computing I. First retake of the midterm 14 December, 2020 1. What is the remainder when we divide 2020

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Introduction to the Theory of Computing I.

First retake of the midterm 14 December, 2020

1. What is the remainder when we divide 20202021 by 1011?

2. We multiply an integer n by 17 and the result gives the remainder 23 when it is divided by 65. What can be the remainder when we divide n by 130?

3. Determine the parametric equations of the line which is obtained as the intersection of the planes given by the equations x+ 3y+ 2z = 7 and 4x+ 6y + 5z = 10.

4. Let V be the set of those vectors in R4 whose coordinates (ordered by position) form an arithmetic progression, i.e. the difference of the consecutive coordinates is the same (if we subtract the coordinate with the smaller index from the coordinate with the greater index). Decide if V is a subspace of R4 or not.

5. The vectors a, b and c are linearly independent in Rn. Does this imply that the vectors 2a, a+b and a+c are linearly independent?

6*. Determine all the integers 1 ≤ a ≤ 100 for which a21 ≡1 (mod 100) holds.

You have 90 minutes to work on the problems and extra 25 minutes to scan and upload your solutions. Each problem is worth 10 points. To obtain a signature you have to achieve at least 24 points.

The details of the solutions must be explained, giving the result only is not worth any points. The problem marked with an * is supposed to be more difficult.

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