781samplemidterm

781samplemidterm - x 8 (mod 21) (d) x 2 22 (mod 5) (e) x 8...

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1. Find the gcd of the pair of integers (399,119). 2. Find all possible integer solutions ( x, y ) to the equation 399 x + 119 y = gcd(399 , 119) . 3. Does the congruence 119 x 14 (mod 399) have a solution? If not, why not. If so, provide at least one solution mod 399. 4. Compute φ (225). 5. Do there exist natural numbers n and m such that φ ( mn ) 6 = φ ( n ) φ ( m )? Explain why or give a counterexample. 6. Find the smallest integer N such that φ ( n ) 5 for all n N . 7. Find TWO solutions n , where n is a positive integer, to the equation φ ( n ) = n/ 3 8. In this question, you ONLY need to determine whether or not the system of equations has at least one solution. If yes, mark “Y”, if no, mark “N.” (a) x 1 (mod 6) x ≡ - 1 (mod 18) (b) 2 x 1 (mod 1234567891011) (c) x 3 (mod 29) x 5 (mod 47) (d) x 22 1 (mod 23) (e) x 2 4 (mod 7) (f) x 2 17 (mod 63) (g) x 3 + x + 57 0 (mod 125) 9. In this question, you ONLY need to determine the NUMBER of solu- tions mod m for each given modulus m . You do not need to determine the actual solutions. (a) 14 x 18 (mod 22)
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(b) 5 x 2718281828 (mod 3141592653) (c) x 7 (mod 33)
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Unformatted text preview: x 8 (mod 21) (d) x 2 22 (mod 5) (e) x 8 1 (mod 20) (f) x 6 + 5 x 4 + 2 x + 1 0 (mod 243) 10. Find all primitive elements in ( Z / 11 Z ) 11. Determine the number of primitive elements in ( Z /m Z ) for the fol-lowing moduli m : (a) m=23 (b) m=49 (c) m=27 (d) m=132 12. In the integers mod 17, 3 is a primitive root. Use the following table of indices to solve the subsequent equations for all x mod 17. (That is, i ( a ) denotes the power of 3 needed to give the residue a mod 17) a (mod 17) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 i(a) (mod 16) 16 14 1 12 5 15 11 10 2 3 7 13 4 9 6 8 (a) x 4 1 (mod 17) (b) 7 x 11 + 5 0 (mod 17) 13. Show that in ( Z /p Z ) , the number of quadratic residues mod p is always equal to the number of quadratic non-residues mod p . 14. Find the number of reduced residues a mod m such that a m-1 1 (mod m ) 15. Show that x 8 16 (mod p ) always has a solution....
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781samplemidterm - x 8 (mod 21) (d) x 2 22 (mod 5) (e) x 8...

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