tut04sols - The University of Sydney MATH2068 Number Theory...

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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 Tutorial 4 1. Solve the following system of simultaneous congruences. 3 x ≡ 1 (mod 7) 2 x ≡ 10 (mod 16) 5 x ≡ 1 (mod 18) Solution. The congruence 3 x ≡ 1 (mod 7) is equivalent to 3 x ≡ - 6 (mod 7), or x ≡ - 2 (mod 7). So x = 7 k- 2 for some k ∈ Z . In the second congruence above we should first divide everything by 2, to convert it to x ≡ 5 (mod 8). This gives 7 k- 2 ≡ 5 (mod 8), and after moving the 2 to the right we find that 7 cancels nicely, giving k ≡ 1 (mod 8), or k = 1 + 8 l for some l ∈ Z . Thus x = 7 k- 2 = 7(1+8 l )- 2 = 5+56 l for some l . Putting this in the last congruence gives 5(5 + 56 l ) ≡ 1 (mod 18), or 5(5 + 2 l ) ≡ 1 (mod 18). This rearranges to 10 l ≡ - 24 (mod 18), or (dividing everything by 2) 5 l ≡ - 12 (mod 9). Or- 4 l ≡ - 12 (mod 9). Cancelling 4 (OK since gcd(4 , 9) = 1) gives l ≡ 3 (mod 9). Thus l = 3 + 9 m for some m ∈ Z , and x = 5 + 56(3 + 9 m ) = 173 + 504 m for some m ∈ Z . So the solution is x ≡ 173 (mod 504). 2. Find the residue of 2 2007 modulo 385. Solution. By Fermat’s Little Theorem, 2 4 ≡ 1 (mod 5), and hence 2 4 k ≡ 1 k ≡ 1 (mod 5) for all k . So 2 2004 ≡ 1 (mod 5), and it follows that 2 2007 = 2 2004 2 3 ≡ 2 3 (mod 5). Similarly, 2 6 ≡ 1 (mod 7) (by Fermat) gives 2 2004 ≡ 1 mod 7 and 2 2007 ≡ 2 3 (mod 7). Since 5 and 7 are coprime it follows at once that 2 2007 ≡ 2 3 ≡ 8 (mod 35). By Fermat’s Little Theorem again, 2 10 ≡ 1 (mod 11), whence it follows that 2 2000 ≡ 1 (mod 11), and 2 2007 ≡ 2 7 (mod 11). But 2 7 = 128 ≡ 7 (mod 11); so our remaining task is to find k ∈ { , 1 , 2 , . . . , 384 } having residue 8 mod 35 and residue 7 mod 11. The first of these conditions gives k = 8 + 35 s for some integer s, and then the second condition gives 8 + 35 s ≡ 7 (mod 11)....
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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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tut04sols - The University of Sydney MATH2068 Number Theory...

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