Use the algorithm in Section 10.1 to find the GCD of the following pairs of integers: (a) 12 and

Use the algorithm in Section 10.1 to find the GCD of the...

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor of two integers is the largest positive integer that divides both numbers without leaving a remainder. This concept is fundamental in number theory and is used in simplifying fractions, solving Diophantine equations, and understanding properties of integers such as coprimality.

Euclidean Algorithm

The Euclidean algorithm is an efficient method for computing the GCD of two integers by iteratively applying the division algorithm. At each step, the larger number is replaced by the remainder when it is divided by the smaller number. This process continues until the remainder is zero, at which point the last nonzero remainder is the GCD.

Transcript

00:01 In this problem, we're going to be using the euclidean algorithm to determine the gcd of six different pairs of integers.

00:10 So how does this work? well, let's say we're given the gcd of a and b.

00:20 Well, here's how the algorithm works.

00:23 We start off with a on the left, and that equals a quotient times b plus a remainder on the right.

00:36 Then we have b on the left now, where we have a new quotient times the remainder in the first line plus a new remainder.

00:50 And this process continues on.

00:53 As you can see here, we have a third quotient, and now the second remainder is here, and now a third remainder is here.

01:01 And this process repeats until the remainder on any line becomes zero.

01:11 So when this number here is equal to zero, then we can say that the greatest common divisor is the remainder of row above that one.

01:24 So it may look confusing like this, but when we're given an example, it'll be much clearer.

01:30 So let's do gcd of 12 comma 18, and this is a...

01:40 Oops, one second.

01:43 This is not good.

01:51 I was writing out of bounds.

01:52 Okay.

01:53 So part a, gcd of 12 and 18.

01:56 So we start with 18 on the left, since this is bigger, equals...

02:02 Well, we can do 1 times 12.

02:05 Any more would be greater than 18, plus a remainder of 6.

02:10 So then we take 12, we put on the left, and we're going to have 6.

02:15 Well, 6 times 2 is 12, so i'll put it to here.

02:19 And notice there is no remainder.

02:21 So this becomes zero.

02:23 Since this is zero, i now can look up here, and i can say that the greatest common divisor is actually 6.

02:31 Okay.

02:34 So that's part a.

02:36 Now we have part b, gcd of 111 and 201.

02:45 Okay.

02:47 Start with the bigger number on the left, equals 1 times 111, plus...

02:56 Let's do some math here.

02:59 Make sure i don't get anything wrong.

03:02 Zero, 1, 9, 0.

03:08 So remainder 90.

03:10 Then we write 111 on the left, equals 90, again, times 1.

03:17 And then we can see that the remainder is 21, because plus 10, then plus 10, plus 1.

03:24 So then we have 90 on the left, equals 21.

03:29 And i'm going to say 4, because 21 times 4 is 84.

03:34 So then the remainder is 6.

03:36 So then we have 21 equals 6 times 3.

03:41 So 6 times 3 is 18.

03:43 So the remainder is 3.

03:46 Now we have 6 equals 3 times 2, plus a remainder of zero.

03:54 So i have a remainder of zero, which means i can stop here and look at the number above.

03:58 So the remainder is 3.

04:00 I'm sorry, the greatest common divisor is 3.

04:03 So that's part b.

04:05 Now we can just divide these up, make it look a little nicer.

04:12 So on the left, now let's do part c.

04:17 C, okay.

04:22 Gcd of 1 ,001, 1331.

04:31 Okay.

04:32 Bigger number on the left, equals 1 times 1 ,001, plus...

04:40 I'm going to say that's 330 on the right, right? 331.

04:48 So then you have 1 ,001 equals 3.

04:53 That's not going over, right? 990.

04:55 So 3 times 330.

04:58 So that's going to be 990.

05:00 So as a remainder, you're going to have 11, i believe.

05:05 990 plus 11.

05:10 Yeah.

05:12 So now we have 330 on the left, equals 11 times...

05:19 So 11 times 3 is 33, times 10 is 330.

05:24 So we're going to have 30 here, and then no remainder.

05:30 And we can stop here because this is so the gcd of these two numbers is 11.

05:40 All right.

05:41 Moving on to part d.

05:43 We have gcd of 1 ,2345, or 12 ,345, and 54 ,321...

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