MATH 406: Homework 18 - Divisibility Tests and Checksums, Assignments of Number Theory

Download MATH 406: Homework 18 - Divisibility Tests and Checksums and more Assignments Number Theory in PDF only on Docsity! MATH 406: Homework 18 Due Tuesday 5/5/2009 Main Concepts • Divisibility Tests • Checksum types. • Single-Digit Errors. 1. Using the tests proven in class, determine whether: (a) 234812501 is divisible by 3. (b) 3817456321 is divisible by 3. (c) 982734521 is divisible by 11. (d) 72403727 is divisible by 11. 2. We proved in class that a number is divisible by 3 if and only if the sum of its digits is divisible by 3. Prove the same for 9. 3. Suppose n = akak−1...a1a0 is a number, where each ai is a digit. Define n ′ = akak−1...a1−2a0. For example, if n = 32587 then n′ = 3258− 2(7) = 3258 − 14 = 3244. (a) Prove that 7 | n if and only if 7 | n′. (b) Use this to prove whether 46075148 and/or 2277887 are divisible by 7 or not, reducing each until it is below 100. 4. Write down the title and ISBN of your favorite book and show verification that it is a valid ISBN. 5. Prove that the UPC Bar Code checksum method can detect any single-digit error. Problems below this point will not be collected. 6. Prove that the Electronic Funds Transfer Routing Number checksum method can detect any single-digit error. 7. Categorize, with proof, the types of transpositions which can and cannot be detected by the Airline Ticket checksum.