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Introduction to Computer Science I. Second Repeat of the First Midterm Test 2017. December 18.

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Introduction to Computer Science I.

Second Repeat of the First Midterm Test 2017. December 18.

1. How many integers x are there between 1 and 2017 for which it holds that 92x−1 andx give the same remainder when divided by 399?

2. Let pbe a positive prime number different from 3 and a be an integer not divisible neither by 3 nor by p. Show that in this case

a6p−6 ≡1 (mod 9p)

3. Let n = 123456. Use the algorithm we learnt to determine the g.c.d.

of 12n+ 6 and 9n+ 4.

4. The system of equations of the line e is x+35 = y+19 =z, and of the line f is x4 = y+86 , z = 7. Determine the system of equations of the normal transversal ofeand f, that is, of the line nwhich intersects both eand f perpendicularly.

5. Let a= (1,2,4)T, b= (0,1,2)T and c= (0,0,1)T be vectors in R3. a) Do the vectors a, b, cform a generating system in R3?

b) Do the vectors a,3a+b,6a+ 2b+cform a generating system inR3? 6. Determine whether the vectors u = (4,3,8,1)T, v = (2,0,4,0)T and

w= (3,5,6,2)T inR4 are linearly independent or not.

The full solution of each problem is worth 10 points. Show all your work!

Results without proper justification or work shown deserve no credit.

Notes and calculators (and similar devices) cannot be used.

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