A1_2008sols - The University of Sydney MATH2068 Number Theory and Cryptography(http/www.maths.usyd.edu.au/u/UG/IM/MATH2068 Semester 2 2008 Lecturer

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Unformatted text preview: The University of Sydney MATH2068 Number Theory and Cryptography (http://www.maths.usyd.edu.au/u/UG/IM/MATH2068/) Semester 2, 2008 Lecturer: R. Howlett Assignment 1 1. Determine the number of positive integers that are divisors of 15!, and their sum. Solution. 15! = (3 5) (2 7) 13 (2 2 3) 11 (2 5) 3 2 2 3 7 (2 3) 5 2 2 3 2 = 2 11 3 6 5 3 7 2 11 13. As proved in lectures, if the prime factorization of the number n is p k 1 1 p k 2 2 p k r r then the values of the number of divisors function and the sum of divisors function are given by ( n ) = Q r i =1 ( k i + 1) and ( n ) = Q r i =1 p k i +1 i- 1 p i- 1 . Thus we see that (15!) = 12 7 4 3 2 2 = 4032 , and (15!) = 2 12- 1 1 3 7- 1 2 5 4- 1 4 7 3- 1 6 11 2- 1 10 13 2- 1 12 = 4095 1093 156 57 12 14 = 6686252969760 . 2. ( i ) Use Fermats Little Theorem to show that 2 2004 1 (mod 13), and hence determine the residue of 2 2008 modulo 13. ( ii ) Find the residue of 2 2008 modulo 11 and the residue of 2 2008 modulo 7. ( iii ) Noting that 1001 = 7 11 13, use your answers to the first two parts to find the residue of 2 2008 modulo 1001. Solution. Fermats Little Theorem says that a p- 1 1 (mod p ) if p is prime and p- a . So 2 12 1 (mod 13). So (2 12 ) 167 1 167 1 (mod 13); that is, 2 2004 1 (mod 13). Hence 2 2008 = 2 2004 2 4 1 2 4 = 16 3 (mod 13). Hence 3 is the residue of 2 2008 modulo 13. By Fermat, 2 10 1 (mod 11), and so 2 2000 = (2 10 ) 200 1 200 1 (mod 11). So 2 2008 1 2 8 = 256 3 (mod 11). Similarly, 2 6 1 (mod 7); so 2 2004 1 (mod 7) and 2 2008 2 4 2 (mod 7). So the residues of 2 2008 modulo 11 and 7 are 3 and 2 respectively....
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This note was uploaded on 09/12/2009 for the course MATH 2068 taught by Professor Howlett during the One '08 term at University of Sydney.

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A1_2008sols - The University of Sydney MATH2068 Number Theory and Cryptography(http/www.maths.usyd.edu.au/u/UG/IM/MATH2068 Semester 2 2008 Lecturer

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