# w6 - 1 5(b Show that 2 has no multiplicative inverse mod 6...

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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 iﬀ 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....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online