12. [1.67/8.37 Points] DETAILS Find 55 307 MY NOTES EPPDISCMATH5 8.4.018. (mod 721) using techniques described in Example 8.4.4 and Example 8.4.5. First compute values of 55k mod 721 for k = 1, 2, 4, 8, 256. For instance, 5532 mod 721 = 614 By Corollary 8.4.4 , if k and m are positive integers, then (55k · 55m) mod 721 = (55k mod 721) · 307 25632+ 16 + 2 + 1 Given that 307 = 256 + 32 + 16 + 2 + 1, 55 Complete the calculations to conclude that 55 mod 721 = 55 mod 721. 307 mod 721 = Need Help? Read It Submit Answer PREVIOUS ANSWERS ASK YOUR TEACHER PRACTICE ANOTHER × and 55256 mod 721 = 460 mod 721

Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .