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 ...continues

Equality of gcd's

Show that if gcd(a, b) = 1, then gcd(ac, b) =gcd(b, c).

The Euler function

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Euler function

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Euler Function

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Euler Function

(See attached file for full problem description) --- We consider the special case when m=3 and n=4. (a) Write down the correspondence between numbers in and pairs of integers in given by the function f. In other words, write out the 12 values f(a) where . (b) Fore each value you computed above, circle the equations ...continues

Euler Function

(See attached file for full problem description) --- We consider the special case when m=3 and n=5. (a) Find the explicit function from the Chinese Remainder Theorem Chapter summary. (Recall that g is the inverse function of f.) (b) Write down all ordered pairs (a,b) Є . (c) Compute g(a,b) for each ordered pair i ...continues

Primitive roots

(See attached file for full problem description with all symbols) Suppose d and n are integers greater than 1 such that . If a is an integer relatively prime to n, show that .

Primitive Roots

(See attached file for full problem description with all symbols) --- Suppose that n is odd and a is a primitive root modulo n. (a) Show that there exists and integer b such that and . (b) Show that b is a primitive root modulo 2n.

Primitive Roots

(See attached file for full problem description) --- Assume r and s are relatively prime positive integers and that n=rs. Let and assume that gcd(a,n)=1. Prove: (a) (b)