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Adaptation course in DM (Number theory) - HW4
This is a test. It goes from simple to less simple problems. The questions with open answer may not give you points. It is ok, just compare your answer with mine and, then, we discuss. Use Lecture 4-5. Try yourself.
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Divisibility
1. Decide whether each of the statements below are true or false.
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1 point
4 | 20
20 | 4
0 | 5
5 | 0
Required
2. Find the remainder when 3^123 is divided by 7. Use the following properties from your lectures.
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3. Let f(x) = 13x^3-5x^2+14x-10. Compute the least positive residue of f(12) mod 7.
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4. How many remainder classes modulo 5 are there?
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5. Use your answer to the previous question to prove that 5 divides n(n^4- 1) for any integer n >=0.
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6. Find great common divisor of 242 and 165 using the formula from the proof of lemma 5.1(see image) or Euclidean algorithm.
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7. Calculate the great common divisor of (5^10 − 1) and (5^ 4-1).
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Solving Congruences
8. Let us start from the Bezout identies. Use the following theorem(and its corollary) to answer which of the following linear equations has a an integer solutions.
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6x + 10y = 1
6x+5y = 1
6x + 10y = 32.
3x + 9y = 26
Required
9. Solve the following linear equation 6x + 5y = 3.
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Your answer
10. Use the previous two questions to answer which of the following congruences has a solution.
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7x ≡ 12 (mod 13)
6x ≡ 1 (mod 10)
14x ≡ 12 (mod 24)
7x ≡ 6 (mod 12)
Required
11. Solve the following congruence 7x ≡ 12 (mod 13).
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12. Which of the following statements are true?
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Diophantine equation 3x+2y =1 have unique integer solution because 3 and 2 are coprime.
Diophantine equation identity 3x+2y =1 have multiple integer solutions because 3 and 2 are coprime.
Congruence 12x ≡ 3 (mod 9) have no solution because 3 and 9 are not coprime.
Congruence 12x ≡ 3 (mod 9) have multiple solution because 3 and 9 are not coprime.
Required
13. Euler's Phi function phi(m), is the total number of elements in {0, 1, 2,...m} that are relatively prime to m. Use the following properties: phi(ab) = phi(a) phi(b) if a, b are coprime; phi(a^k) = a^k - a^(k-1) to calculate the phi(210).
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14*. Find all n >= 0 for which 3^n + 4^n = 0 (mod 7).
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15*. Let F(n) be the Fibonacci numbers, defined by F(0)=0, F(1)=1. F(n)=F(n-1)+F(n-2). Find if possible the solution to F(100)x + F(99)y = 1. There is no need to calculate the Fibonacci numbers.
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