Use the Euclidean algorithm to find a) gcd(1,5) . b) gcd(100,101) c) gcd(123,277) .

Use the Euclidean algorithm to find a) gcd(1,5) . b)...

Key Concepts

Greatest Common Divisor (gcd)

The greatest common divisor is a fundamental concept in number theory that represents the largest positive integer that divides two numbers without leaving a remainder. It is crucial for simplifying fractions, solving Diophantine equations, and understanding divisibility properties between integers.

Division Algorithm

The division algorithm is a principle that states for any two integers, a dividend and a non-zero divisor, there exist unique integers – a quotient and a remainder – such that the dividend equals the divisor multiplied by the quotient plus the remainder. This concept underpins many algorithms in number theory, including the Euclidean algorithm.

Euclidean Algorithm

The Euclidean algorithm is an efficient method for computing the greatest common divisor of two numbers by repeatedly applying the division algorithm. By iterating on the pair of numbers with the larger replaced by the remainder from dividing the larger by the smaller, the algorithm eventually reaches a stage where the remainder is zero. At that point, the non-zero divisor is the gcd.

Remainder Calculation

Remainder calculation is an operational step in the division process where the remainder is found after dividing one integer by another. It is essential for the Euclidean algorithm as it determines the next step in the iterative process, gradually reducing the problem size until the gcd is found.

Transcript

00:01 We're asked to use the euclidean algorithm to find the greatest common divisors of a pair of numbers.

00:09 In part a, we're asked to find the greatest common divisor of 1 and 5.

00:17 Well, we see that 5 is the larger of the two numbers.

00:23 So 5 is equal to 1 times 5, and therefore it follows that the greatest common divisor of 1 and 5 is equal to now in part b, we're asked to find the greatest common divisor of 100 and 101.

01:01 So 101 is the greater of these two numbers.

01:04 101 is equal to 100 times 1 plus 1.

01:13 And we have that 100 is the greater of 100 and 1.

01:18 And so this is equal to 1 times 100.

01:29 So we see the last non -zero.

01:32 Remainder is 1.

01:37 So it follows that the greatest common divisor of 100 and 101 is 1.

01:50 In part c, we're asked to find the greatest common divisor of 123 and 277.

02:01 The greater of these two numbers is 277.

02:05 So dividing 277 by 123, we get 277 is equal to 123 times 2.

02:14 This is 246 plus 11, sorry, plus 31.

02:33 And then we'll divide 123 by 31, since 31 is a non -zero remainder.

02:41 So 123 is equal to 31 times 3.

02:47 This is 93 plus 30.

02:52 This is another non -zero remainder.

02:55 So we'll divide 31 by 30.

02:58 And we get 31 is equal to 30 times 1 plus 1, and 1 is a non -zero remainder.

03:08 So finally we'll divide 30 by 1, and 30 is equal to 30 times 1.

03:14 Or 1 times 30, i guess, to be consistent.

03:19 So it follows the last non -zero remainder is 1, and therefore that the greatest common divisor of 277 and 123 is 1.

03:36 In part d, we're asked to find the greatest common divisor of 1 ,529 and 14 ,039.

03:49 Now, we have that 14 ,029 is the greater of these two, so we'll divide it by 1 ,529.

03:59 This is equal to 1 ,529, and then multiply this by 9.

04:07 So we get 9 ,000 plus 4 ,500.

04:13 So that's 13 ,500, 13 ,500 plus 180 is 13 ,680 plus 81, and then we have the 14 ,038 minus 13 ,000, whatever i said.

04:48 Said is 277.

04:56 Now we see that this is a non -zero remainder, so we'll divide 1 ,529 by 277.

05:06 So this is equal to 277 times 5.

05:15 So that gives us 200 times 5 is 1 ,000 plus 350 is 1350, plus 35 is 1385, and then subtracting that from 1529, we get 144.

05:36 Now, this is another non -zero remainder, so we'll divide 277 by 144.

05:45 This is equal to 144 times 1 plus, and then 277 minus 144, which is 133.

05:58 This is another non -zero remainder, so we'll divide 144 by 133.

06:06 So this is 133 times 1 plus 11.

06:17 This is another non -zero remainder, so we'll divide 133 by 11.

06:22 This is 11 times 12...

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