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

GCD (Greatest Common Divisor) - dCode

Tag(s) : Arithmetics

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**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 (prefered method for calculators)

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

**GCD Method 4**: knowing the GCD, use the formula GCD(a, b) = a * b / LCM(a, b)

__Example:__ The LCM (least common multiple) of 10 and 12 is 60, so GCD(10, 12) = 10 * 12 / 60 = 2

**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

To simplify a fraction, it is possible to divide the numerator and demonimator by their GCD to obtain an irreducible fraction.

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 $

HCF stands for highest common factor, it is exactly the same thing as GCD.

The program ignores 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. By convention, only the positive value is given. $$ GCD(a,b) = GCD(-a,b) = GCD(a,-b) = GCD(-a,-b) $$

__Example:__ In this second case, for all solution, the opposite is valid: GCD(6,9) = GCD(-6,9) = GCD(6,-9) = GCD(-6,-9) = 3 (ou -3).

An alternative method to euclidean divisions using successive subtractions based on the property $$ gcd(a,b) = gcd(b,a) = gcd(b,a-b) = gcd(a,b-a) $$

__Example:__ GCD(12, 10) = GCD(10, 12-10=2) = GCD(2, 10-2=8) = GCD(8, 8-2=6) = GCD(6, 8-6=2) = GCD(6, 6-2=4) = GCD(4, 6-4=2) = GCD(4, 4-2=2) = GCD(2, 2) = 2.

Use the formula $ GCD(a,b) = (a \times b) / LCM(a, b) $

with $ a \times b $ the product of the 2 numbers and LCM their least common multiple

`// 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);

}

// Python

def gcd(a, b):

while b!=0:

a,b=b,a%b

return a

Using prime factor decomposition

$$ b = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_n^{a_n} $$

$$ c = q_1^{b_1} \times q_2^{b_2} \times \cdots \times q_m^{b_m} $$

As GCD(b,c)=1, no factor $ p $ is equal to any factor $ q $. However $ GCD(a,b) $ is a product of factors $ p $ and $ GCD(a,c) $ is a product of factors $ q $ and $ GCD(a, b \times c) $ is a product of factors $ p $ and $ q $. So $ GCD(a, b \times c) = GCD(a,b) \times GCD(a,c) $

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

For Casio

// GCD Finder

"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 TI (82,83,84,89)`Input "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

The GCD is a common divisor (the greatest) of the 2 numbers, which is a smaller number having both numbers for multiples.

The LCM is a common multiple (the lowest) of the 2 numbers, which is a larger number having both numbers for divisors.

The CGD and the LCM are linked by the formula: $$ GCD(a, b) = \frac{ a \times b }{ LCM(a, b) } $$

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*GCD (Greatest Common Divisor)* on dCode.fr [online website], retrieved on 2022-09-30,

- GCD of 2 or more numbers Calculator
- LCM Calculator
- List of Divisors
- What is the GCD? (Definition)
- How to calculate the GCD? (Algorithm)
- How to find the GCD with multiple numbers? (GCD of 3 numbers or more)
- Why calculate GCD of numerator and denominator?
- What is the definition of two relatively prime numbers?
- What is the différence between GCD and HCF?
- How to calculate GCD with negative integers?
- How to calculate GCD with subtractions?
- How to calculate GCD of 2 numbers knowing their product and their LCM Calculator?
- How to code a GCD algorithm?
- How to demonstrate that if GCD(b,c)=1, then GCD(a,b*c) = GCD(a,b).GCD(a,c)?
- How to calculate GCD with a calculator (TI or Casio)?
- What is the difference between GCD and LCM Calculator?

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