New PDF release: Mental Maths Book 1 (Bk.1)

By Anne Foster

ISBN-10: 0721709621

ISBN-13: 9780721709628

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Sample text

Repeate d applicatio n o f this reductio n metho d mus t eventuall y reac h a proble m o f thi s for m in which a divides b. Bu t ax = c mod qa has a solution i f and onl y if a divides c, in which case the most genera l solution is x = f mo d | . ) Express thi s metho d o f solvin g ax = c mod b a s a forma l algo rithm. Computations. 9. Implemen t th e augmente d Euclidea n algorith m o n a compute r and see fo r yoursel f ho w wel l i t work s eve n wit h number s tha t hav e many, man y digits .

Bu t i f a prime facto r p o f 0(c ) i s foun d fo r whic h a^ c^p = 1 mod c , on e ca n then begi n tryin g exponent s whic h ar e 0(c)/ p divide d b y on e o f its prime factors , an d s o forth. Eventuall y (an d usuall y rathe r soon ) on e will find a factor / o f 0(c) fo r which a? = 1 mod c but a^ p ^ 1 mod c for eac h prim e facto r p o f / . Thi s / i s the orde r o f a mod c . Traditionally, th e fundamenta l fac t o f elementar y numbe r the ory expresse d b y th e corollar y ha s th e unsatisfactor y nam e "Euler' s generalization o f Fermat' s theorem.

Proposition. This problem has a solution b if and only if a is relatively prime to c. When it has a solution, every solution is a multiple of the smallest solution. Proof. I f the problem has a solution, say ab = 1 mod c , then a b+sc = 1 + tc fo r som e number s s an d t. Sinc e a b i s a multipl e o f a , th e equation a b + sc = 1 + tc show s tha t 0 = 1 mod [a , c] whic h mean s that a an d c are relativel y prime . For th e proo f o f the converse , assum e a and c are relatively prim e and conside r th e firs t c + 1 powers a, a 2 , a 3 , .

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Mental Maths Book 1 (Bk.1) by Anne Foster

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