### Bank Account Check Digit (Congruences modulo 10) The identification number for a bank printed on a check consists of eight digits x1 . . . x8, followed by a ninth check digit x9, with x9 ≡ 7x1 +3x2 + 9x3 + 7x4 + 3x5 + 9x6 + 7x7 + 3x8 (mod 10). (a) What is the check digit following the eight-digit identification number 00185403 for a bank? (b) Will this scheme detect a change in one of the digits? Prove your answer. (c) Will this scheme detect a change the transposition of two consecutive, but distinct digits? Prove your answer.

The identification number for a bank printed on a check consists of eight digits x1 . . . x8, followed by a ninth check digit x9, with x9 ≡ 7x1 +3x2 + 9x3 + 7x4 + 3x5 + 9x6 + 7x7 + 3x8 (mod 10). (a) What is the check digit following the eight-digit identification number 00185403 for a bank? (b) Will this scheme d