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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

If c a c b ( mod n ) and d = ( c , n ) where n = d m , prove that a b ( mod n ) .

To determine

To prove: If cacb(modn) and d=(c,n), where n=dm, then ab(modm).

Explanation

Given information:

cacb(modn) and d=(c,n), where n=dm.

Formula used:

i) Definition: Congruence Modulo n

Let n be a positive integer, n>1. For integers x and y, x is congruent to y modulo n, if and only if xy is a multiple of n. We write xy(modn) to indicate that x is congruent to y modulo n.

ii) Definition: An integer d is a greatest common divisor of a and b, if all these conditions are satisfied:

1. d is a positive integer.

2. d|a and d|b.

3. c|a and c|b imply c|d.

iii) If d=(a,b),a=a0d, and b=b0d, then (a0,b0)=1.

iv) Theorem: Cancellation Law:

If axay(modn) and (a,n)=1, then xy(modn).

Proof:

Let cacb(modn) and d=(c,n), where n=dm

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