Question

Use the Euclidean algorithm to compute gcd(520038035,622186145) = gcd(991966975,408146589) =

A. Not enough information
B. Mathematical proof
C. Number theory principles
D. Step-by-step computation

Answer 1

Answer:

  gcd(520038035,622186145) = 5

  gcd(991966975,408146589) = 1

Step-by-step explanation:

You want the GCDs of the given pairs of numbers, using the Euclidean algorithm.

Euclidean algorithm

The algorithm works like this:

1. Find the remainder of the division of the larger number by the smaller.
2. If the remainder is 0, the smaller is the GCD. (Done)
3. Replace the larger number by the remainder and repeat from step 1.

GCD(520038035,622186145)

The first four attachments show the computation of the GCD. Successive remainders are ...

  102148110, 9297485, 9173260, 124225, 104835, 19390, 7885, 3620,

  645, 395, 250, 145, 105, 40, 25, 15, 10, 5, 0

The GCD is 5.

GCD(991966975,408146589)

Simple divisibility checks tell you these numbers do not have factors of 3, 5, or 9 in common. This suggests executing the algorithm will be just as tedious as for the previous pair of numbers.

The calculator's GCD function tells us the GCD of these two numbers is 1. Their prime factors confirm this result, as shown in the 5th attachment.

The GCD is 1.

Answer 2

Final answer:

The Euclidean algorithm is used to find the greatest common divisor (gcd) of two numbers through a series of division steps, taking the remainder of the previous division as the new divisor until a remainder of zero is reached. To compute gcd(520038035,622186145) and gcd(991966975,408146589), the same iterative process is applied to each pair of numbers.

Explanation:

The Euclidean algorithm is a method used to find the greatest common divisor (gcd) of two integers. To compute gcd(520038035,622186145) using the Euclidean algorithm, you perform a series of division steps where, in each step, you take the remainder of the previous division as the new divisor until you reach a remainder of zero. The non-zero remainder just before zero is the gcd of the two numbers.

Step-by-Step Computation:

1. Divide 622186145 by 520038035 and get the remainder. Let's call this remainder r1.
2. Now divide 520038035 by r1 and get the new remainder. Call this r2.
3. Continue this process, dividing the last non-zero remainder by the new remainder each time, until you reach a remainder of zero.
4. The gcd is the last non-zero remainder.

Similarly, to find gcd(991966975,408146589), you would follow the exact same division steps with these numbers. The Euclidean algorithm is a example of Number Theory Principles at work. It is not necessary to use any additional information or mathematical proofs as the algorithm itself suffices to determine the gcd.