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Asked Nov 24, 2019
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solve part b

a: Assuming a > b > 0 prove gcd(a, b) = gcd(b, r) where r is the remainder when a is divided by
b.
b: Use the Euclid's Algorithm to find gcd(44, 104), and then find two integers æ and y that safisfy
44.x104y=
gcd(44, 104)
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a: Assuming a > b > 0 prove gcd(a, b) = gcd(b, r) where r is the remainder when a is divided by b. b: Use the Euclid's Algorithm to find gcd(44, 104), and then find two integers æ and y that safisfy 44.x104y= gcd(44, 104)

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Expert Answer

Step 1

Let the number a and b can be written as a = b q + r where r lies between 0 and a.

 

Now we have to prove that gcd (a, b) = gcd (b, r)

 

 

GCD of two numbers a and b is defined as the biggest number which divides both a as well as b, denoted by gcd (a, b).

 

Now a = b q + r, then d is the common divisor between a and b,

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(r = a-bq) a-ba) and thus -b

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Step 2

Thus d is the common divisor of r and b,

 

Now d is the common divisor of r and b. Then:

 

...
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dand dthus(bq+r) (abq+r)

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