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# s7 - Math 55 Worksheet Adapted from worksheets by Rob Bayer...

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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) x = - 4 Linear Congruences and the CRT 1. Solve the system of congruences x 3 ( mod 4) x 2 ( mod 5) x 5 ( mod 7) Solution: x 47 mod 140. 2. Solve the system of congruences 2 x 1 ( mod 3) 3 x 2 ( mod 5) 4 x 1 ( mod 7) Solution: To start this one, notice what happens if we multiply both sides of the first equation by 2! Since 2 · 2 1 mod 3, the first equation is the same as x 2 mod 3. Similarly, we take care of the other equations. Here’s a website which talks about the CRT and has a calculator to help you solve problems like this: (And plenty of other good resources too! http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/chinese3.html) 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).

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s7 - Math 55 Worksheet Adapted from worksheets by Rob Bayer...

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