s6 - Math 55 Worksheet Adapted from worksheets by Rob Bayer...

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Math 55 Worksheet Adapted from worksheets by Rob Bayer, Summer 2009. Divisibility and Modular Arithmetic 1. Evaluate each of the following: (a) - 17 mod 2 1 (b) 144 mod 7 4 (c) 199 mod 19 9 (d) - 101 div 13 8 - (Not 7)! 2. What is 111 ··· 1 | {z } 1000 1 0 s mod 11111111 | {z } 8 1 0 s ? Let’s try a simpler example to see what’s going on: Instead of 1000 1’s, let’s try just 16: 111 ··· 1 | {z } 16 1 0 s : What do we know about this number? Well let’s just write it down! 1111111111111111 = 1111111100000000 + 11111111 Do you see how I split it up? Into groups of 8 1’s? Do you believe that each term in the above is divisible by 11111111? Now what if we had 24 zeros? 11111111 , 11111111 , 11111111 = 11111111 , 00000000 , 00000000+11111111 , 00000000+11111111? We could write this as 111 ··· 1 | {z } 24 1 0 s = 11111111 · 10 16 + 11111111 · 10 8 + 11111111 . And now each term is clearly di- visible by 111 ··· 1 | {z } 8 1 0 s . So now in general, we have that 111 ··· 1 | {z } 1000 1

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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 Berkeley.

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s6 - Math 55 Worksheet Adapted from worksheets by Rob Bayer...

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