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1. Prove that 3n + 2 and 5n + 3 are relatively prime for every positive integer n . 2. Find d = gcd(574, 252) and solve the equation 572a + 252b = 28 with a, b ∈ Z 3. Show that if n ≥ k ≥ r ≥ 0 we have ! ! ! ! n k n n−r = k r r k−r ! n X n 4. Prove that = 2n , for every n ≥ 1. k k=0 5. Use the Euclidean algorithm to compute the greatest common divisor of 1197 and 14280, and to express it as a linear combination of 1197 and 14280. 6. Show that for n ≥ 1: ! 2n 1 · 3 · 5 · · · (2n − 1) × 22n = n 2 · 4 · 6 · · · 2n 7. Prove that 3a2 − 1, a ∈ Z can never be a perfect square, i.e = b2 , for some integer number b. 8. Show that gcd(2a + 1, 9a + 4) = 1 , ∀a ∈ Z. 9. Define a sequence of integers as follows: f0 = 0, f1 = 1, fn = fn−1 + fn−2 , n≥2 Prove that gcd(fn , fn+1 ) = 1 for every n ≥ 1. 10. If x is any real number, define bxc = the greatest integer less than or equal to x (the floor of x) and dxe the least integer greater than or equal to x (the ceiling of x). Define also {x} = x − bxc the fractional part of x. • What are b−1.1c, b0.99999c, d−1.1e, d1.01e? • Prove that for every x ∈ R b−xc = −dxe. • Whichp of the following equations are true for all positive real numbers x? √ a) bp bxcc = b xc √ b) d bxce = d xe 11. Factor 51, 948 into a product of primes. 12. Prove that n5 − n is divisble by 30 for every integer n. 13. Prove that n, n + 2, n + 4 are all primes if and only if n = 3. n 14. The integer Fn = 22 +1 is called the nth Fermat number. Show that for n = 1, 2, 3, 4 Fn is prime. 15. Show that F5 is not prime. 16. Prove that the equation 3x1 + 5x2 = b has a solution in non-negative integers for b = 0, 3, 5, 6 and all b ≥ 8. 17. Prove a3 ≡ a ( mod 6) for every integer a. 18. Prove that a4 ≡ 1 ( mod 5) for every integer a that is not divisble by 5.