75 lines
2.4 KiB
Markdown
75 lines
2.4 KiB
Markdown
---
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title: Greatest Common Divisor Euclidean
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---
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## Greatest Common Divisor Euclidean
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For this topic you must know about Greatest Common Divisor (GCD) and the MOD operation first.
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#### Greatest Common Divisor (GCD)
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The GCD of two or more integers is the largest integer that divides each of the integers such that their remainder is zero.
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Example-
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GCD of 20, 30 = 10 *(10 is the largest number which divides 20 and 30 with remainder as 0)*
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GCD of 42, 120, 285 = 3 *(3 is the largest number which divides 42, 120 and 285 with remainder as 0)*
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#### "mod" Operation
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The mod operation gives you the remainder when two positive integers are divided.
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We write it as follows-
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`A mod B = R`
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This means, dividing A by B gives you the remainder R, this is different than your division operation which gives you the quotient.
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Example-
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7 mod 2 = 1 *(Dividing 7 by 2 gives the remainder 1)*
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42 mod 7 = 0 *(Dividing 42 by 7 gives the remainder 0)*
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With the above two concepts understood you will easily understand the Euclidean Algorithm.
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### Euclidean Algorithm for Greatest Common Divisor (GCD)
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The Euclidean Algorithm finds the GCD of 2 numbers.
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You will better understand this Algorithm by seeing it in action.
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Assuming you want to calculate the GCD of 1220 and 516, lets apply the Euclidean Algorithm-
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Assuming you want to calculate the GCD of 1220 and 516, lets apply the Euclidean Algorithm-
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Pseudo Code of the Algorithm-
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Step 1: **Let `a, b` be the two numbers**
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Step 2: **`a mod b = R`**
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Step 3: **Let `a = b` and `b = R`**
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Step 4: **Repeat Steps 2 and 3 until `a mod b` is greater than 0**
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Step 5: **GCD = b**
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Step 6: Finish
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Javascript Code to Perform GCD-
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```javascript
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function gcd(a, b) {
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var R;
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while ((a % b) > 0) {
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R = a % b;
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a = b;
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b = R;
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}
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return b;
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}
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```
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Javascript Code to Perform GCD using Recursion-
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```javascript
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function gcd(a, b) {
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if (b == 0)
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return a;
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else
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return gcd(b, (a % b));
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}
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```
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You can also use the Euclidean Algorithm to find GCD of more than two numbers.
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Since, GCD is associative, the following operation is valid- `GCD(a,b,c) == GCD(GCD(a,b), c)`
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Calculate the GCD of the first two numbers, then find GCD of the result and the next number.
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Example- `GCD(203,91,77) == GCD(GCD(203,91),77) == GCD(7, 77) == 7`
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You can find GCD of `n` numbers in the same way.
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