w6 - 1 5 . (b) Show that 2 has no multiplicative inverse...

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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 (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 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 like
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Unformatted text preview: 1 5 . (b) Show that 2 has no multiplicative inverse mod 6. This means that 1 2 has no meaning when working mod 6. 8. Show that a natural number n is divisible by 3 i the sum of the digits is also divisible by 3. The Euclidean Algorithm 1. Find the gcd of each of the following pairs of numbers and write it as a sa + tb for some integers s,t . (a) 21,55 (b) 123, 323 2. Find the multiplicative inverse of 9 mod 20. 3. Solve the congruence 5 x 16( mod 21) 4. * Show that log 2 3 is irrational....
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