3.4e.pptx - The Divisibility Modular Arithmetic Selected...

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The Divisibility & Modular Arithmetic: Selected Exercises
Exercise 10 Division Algorithm : Let a Z , d Z + . !q !r ( 0 r < d a = dq + r ). What are the quotient & remainder when: a)
b)
c)
d)
e)
f)
g)
h)
Copyright © Peter Cappello 2013 2 Remember: remainders are always nonnegative .
Copyright © Peter Cappello 2013 3 Exercise 34 Thm. 4. Let m Z + and a, b Z. a is congruent to b modulo m, denoted a b ( mod m ), k Z ( a = b + km ). ( m | ( a – b ) ) Show: if a b ( mod m ) and c d ( mod m ), where a, b, c, d , m Z with m 2, then a – c b – d ( mod m ).
Copyright © Peter Cappello 2011 4 Exercise 20 Solution Thm 4. Let m Z + . Integers a & b are congruent modulo m k Z , a = b + km. ( m | ( a – b ) ) Show: if a b ( mod m ) and c d (mod m ), where a, b, c, d , m Z with m 2, then a – c b – d ( mod m ).

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