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

GCD (Greatest Common Divisor) - dCode

Tag(s) : Arithmetics

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

**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

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 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. By convention, only the positive value is given. $$ PGCD(a,b) = PGCD(-a,b) = PGCD(a,-b) = PGCD(-a,-b) $$

__Example:__ 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).

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

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

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

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

dCode retains ownership of the "GCD (Greatest Common Divisor)" source code. Except explicit open source licence (indicated Creative Commons / free), the "GCD (Greatest Common Divisor)" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or the "GCD (Greatest Common Divisor)" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "GCD (Greatest Common Divisor)" are not public, same for offline use on PC, tablet, iPhone or Android !

The copy-paste of the page "GCD (Greatest Common Divisor)" or any of its results, is allowed as long as you cite the online source

Reminder : dCode is free to use.

- 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)
- What is the definition of two numbers relatively primes?
- 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 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?

gcd,greatest,highest,common,divisor,algorithm,integer,division,euclidean,euclide,calculate

Source : https://www.dcode.fr/gcd

© 2022 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback