# ws6 - b ( mod n ) 5. Prove that if the last digit of n is...

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Rob Bayer Math 55 Worksheet July 1, 2009 Instructions Introduce yourselves! Despite popular belief, math is in fact a team sport! Find some blackboard space, a piece of chalk, and decide who will be your ﬁrst scribe. Do the problems below, having a diﬀerent person be the scribe for each one. Try to work out the problems as a group, but feel free to ﬂag me down if you run into a wall. Divisibility and Modular Arithmetic 1. Evaluate each of the following: (a) - 17 mod 2 (b) 144 mod 7 (c) 199 mod 19 (d) - 101 div 13 2. What is 111 ··· 1 | {z } 1000 1 0 s mod 11111111 | {z } 8 1 0 s ? 3. Prove that if a b ( mod m ) and c d ( mod m ), then ac bd ( mod m ) 4. Show that if n | m and a b ( mod m ), then a
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Unformatted text preview: b ( mod n ) 5. Prove that if the last digit of n is 3, then n is not a perfect square. 6. Give an example of integers a,k,l,m such that k l ( mod m ), but a k 6 a l ( mod m ) 7. (a) Find a solution to 5 x 1 ( mod 6). Your answer is called a multiplicative inverse of 5, mod 6 because it behaves similar to 1 5 . (b) Show that 2 has no multiplicative inverse mod 6. That is, show that 1 2 has no meaning when working mod 6. 8. Show that a natural number n is divisible by 11 i the alternating sum of its digits is too (ie, rst digit -second + third - fourth + )...
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## This note was uploaded on 08/31/2009 for the course MATH 55 taught by Professor Strain during the Summer '08 term at University of California, Berkeley.

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