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GCD (Greatest Common Divisor)

Tool to compute GCD. The greatest common divisor of two integers is the greatest positive integer which divides simultaneously these two integers.

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GCD (Greatest Common Divisor) -

Tag(s) : Arithmetics, Mathematics

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GCD (Greatest Common Divisor)

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Calculus of GCP of any numbers



List of Divisors

Tool to compute GCD. The greatest common divisor of two integers is the greatest positive integer which divides simultaneously these two integers.

Answers to Questions

How to calculate the GCD? (Algorithm)

GCD Method 1: list divisors of each number and find the greatest common divisor.

Example: GCD of the numbers 10 and 12.
10 has for divisors' list: 1,2,5,10
12 has for divisors' list: 1,2,3,4,6,12
The greatest common divisor (of these lists) is 2 (The largest number in all lists).
So, GCD(10,12) = 2

GCD Method 2: use Euclidean algorithm

Step 1. Make an euclidean division of the largest of the 2 numbers A by the other one B, to find a dividend D and a remainder R. Keep the numbers B and R.

Step 2. Repeat step 1 (with numbers kept, B becomes the new A and R becomes the new B) until the remainder is zero.

Step 3. GCD of A and B is equal to the last non zero remainder.

Example: A=12 and B=10, and (step 1) compute A/B = 12/10 = 1 remainder R=2.
(step 2) 10/2 = 5 remainder 0, the remainder is zero.
The last remainder not null is 2, so GCD(10, 12) = 2.

GCD Method 3: use prime factor decomposition

GCD is the multiplication of common factors (e.g. the product of all numbers presents in all decompositions).

Example: Numbers 10 and 12 which prime decomposition are: 10 = 2 * 5 and 12 = 2 * 2 * 3. The only common factor is 2. So GCD(10,12) = 2

How to find the GCD with multiple numbers? (GCD of 3 numbers or more)

GCD Method 1: list divisors of the numbers and find the greatest common divisor.

Example: Search for the GCD of the numbers 10, 20 and 25.
10 has for divisors 1,2,5,10.
20 has for divisors 1,2,4,5,10,20.
25 has for divisors 1,5,25.
The greatest common divisor is 5.

GCD Method 2: use the formula GCD(a,b,c) = GCD( GCD (a,b) , c )

Example: GCD (10,20) = 10

Example: GCD (10,20,25) = GCD( GCD(10,20), 25) = GCD(10, 25) = 5

GCD Method 3: use prime factor decomposition

Example: 10 = 2 * 5
20 = 2 * 2 * 5
25 = 5 * 5

GCD is the multiplication of common factors

Example: GCD (10,20,25) = 5

What is the definition of two numbers relatively primes?

Two numbers \( a \) and \( b \) are said to be relatively prime if there is no number except \( 1 \) which is both the divisor of \( a \) and \( b \).

Two numbers \( a \) and \( b \) are said to be co-prime if their GCD is \( 1 \): \( gcd(a,b) = 1 \)

How to calculate GCD with negative integers?

The program ignore negative numbers. To be rigorous mathematically, it depends on the definition of PGCD, defined over N*, it is always positive, defined over Z* it can be negative, but it is the same, with a -1 coefficient.

In this second case, for all solution, the opposite is valid: PGCD(6,9) = PGCD(-6,9) = PGCD(6,-9) = PGCD(-6,-9) = 3 (ou -3). By convention, only the positive value is given.

How to calculate GCD with subtractions?

An alternative method to euclidean division uses successive subtractions based on the properties pgcd(a,b) = pgcd(b,a) = pgcd(b,a-b) = pgcd(a,b-a).

Example: PGCD(12, 10) = PGCD(10, 12-10=2) = PGCD(2, 10-2=6) = PGCD(6, 6-2=4) = PGCD(4, 4-2=2) = PGCD(2, 2) = 2.

How to code GCD algorithm?

// JAVASCRIPT
function pgcd(a,b) {
return (b==0)?a:pgcd(b,a%b);
}
// PHP
function pgcd($a,$b) {
return ($b==0)?$a:pgcd($b,$a%$b);
}

How to demonstrate that if GCD(b,c)=1, then GCD(a,b*c) = GCD(a,b).GCD(a,c)

Using prime factor decomposition

$$ b = p_1^{a_1} * p_2^{a_2} * ... * p_n^{a_n} $$

$$ c = q_1^{b_1} * q_2^{b_2} * ... * q_m^{b_m} $$

As GCD(b,c)=1, no factor p is equal to q

However GCD(a,b) = a product of factors p

And GCD(a,c) = a product of factors q

And PGCD(a,b*c) = a product of factors p and q

So PGCD(a,b*c) = PGCD(a,b) * PGCD(a,c)

How to calculate GCD with a calculator (TI or Casio)?

Calculators has generally a function for GCD, else here are programs

For Casio"A=" : ? -> R
"B=" : ? -> Y
I -> U : 0 -> W : 0 -> V : I -> X
While Y <> 0
Int(R/Y) -> Q
U -> Z : W -> U : Z-Q*W -> W
V -> Z : X -> V : Z-Q*X -> X
R -> Z : Y -> R : Z-Q*Y -> Y
WhileEnd
"U=" : U : "V=" : V
"PGCD=" : R

for TIInput "A=", R
Input "B=", Y
I -> U : 0 -> W : 0 -> V : I -> X
While Y <> 0
Int(R/Y) -> Q
U -> Z : W -> U : Z-Q*W -> W
V -> Z : X -> V : Z-Q*X -> X
R -> Z : Y -> R : Z-Q*Y -> Y
End
Disp "U=", U, "V=3, V
Disp "PGCD=", R

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