Math Assignment

Math Assignment

Question1

Use the Euclidean Algorithm to find the GCF for the natural numbers28 and 96. Show both the geometric model and the numerical model.

Answer 1

It is the method for finding the greatest common factor of two integers. Divide the larger number by the smaller number. The division is repeated, using the residue as divisor, until the residue becomes zero. The last non-zero remainder is the greatest common factor of two integers. For example, to find the highest common factor of 99 and 44, divided 44 by 99. This gives 2 with a remainder of 11. Then, 11 is divided between 99, which gives 9 with a remainder of 0. Since 11 is the last nonzero remainder is the greatest common factor of 99 and 44.

Another example would be the following

For example, to find the greatest common divisor of 93164 and 5826 the algorithm generates the following sequence of sections:

Step Operation Meaning

1 93164 divided by 5826 is 15 and left over 5774

February 5826 divided by 5774 is 1 and can spare 5

March 5774 divided by 52 is 111 and spare 2

April 52 divided by 2 is 26 and spare 0

Its computational complexity would be O (log n) divisions for calculating the maximum common divisor of n and m where

n> m

Its implementation in pseudo code would be:

Algorithm recurrently

Function

gcd (a, b)

If b = 0 then:

The result is a

In another case:

The result is gcd (b, a mod b)

Algorithm iteratively

Function

gcd (a, b):

while do the following:

Question 2

Use Fermat's algorithm to find the prime factors of the natural numbers 125.

Answer 2

The 4 variables of 125 are 1, 5, 25, and 125. The element combines of 125 are 1 x 125 and 5 x 25. The correct components of 125 are 1, 5, and 25 or, if the definition you are utilizing prohibits 1, they are 5. furthermore 25. The prime variables of 125 are 5, 5, and 5. Note: There is reiteration of these variables, so if the prime components are constantly recorded in place of the prime factorization, normally just the notable prime elements are recorded.

The main dissimilar prime component (posting every prime figure just once) of 125 is 5. The prime factorization of 125 is 5 x 5 x 5 or, in list shape as such, utilizing types), 53. Note: There can't be normal elements, a most terrific regular variable, or a minimum regular various of a solitary number on the grounds that "normal" alludes to variables or products that two or more numbers have in as a relatable point.

Fermat factorization

This factorization algorithm was developed by Pierre de Fermat. The algorithm is based on the representation of the odd integers as the difference of two squares.

N=a2-b2

This difference can be factored as follows:

A {2} ^ - ^ {2} b = (a + b) (ab)

If N = cd is a factorization of N then

N=a2-b2

C=a+b

D=ab

C+d=2a

A= fracc+d2

Cd= 2b

B= fraccd2

$ C = (\ frac {c ...