w7 - Math 55 Worksheet Adapted from worksheets by Rob...

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Math 55 Worksheet Adapted from worksheets by Rob Bayer, Summer 2009. Warmup 1. Solve the congruence 5 x 16( mod 21) Linear Congruences and the CRT 1. Solve the system of congruences x 3 ( mod 4) x 2 ( mod 5) x 5 ( mod 7) 2. Solve the system of congruences 2 x 1 ( mod 3) 3 x 2 ( mod 5) 4 x 1 ( mod 7) 3. F Suppose we want to find all solutions to x 2 4 ( mod 35) (a) Show that such solution must also be a solution to the system ( x 2 4 ( mod 5) x 2 4 ( mod 7) and vice-versa. (b) Show that, based on the conditions in (a), x 2 or 3 ( mod 5) and that x 2 or 5 ( mod 7). (c) Use the CRT on each of the possibilities from (b) to find all solutions to x 2 4 ( mod 35). 4. Compute the following: (a) 7 130 mod 11, (b) 4 16 mod 10, and (c) 10 4 16 mod 11. 5. F Let p be a prime number and let A = { 1 , 2 , ··· ,p - 1 } . Because all elements of A are relatively prime to p , they all have inverses (mod p ). Let
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This note was uploaded on 02/06/2012 for the course MATHEMATIC 55 taught by Professor Robbayer during the Summer '09 term at University of California, Berkeley.

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w7 - Math 55 Worksheet Adapted from worksheets by Rob...

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