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a) Define the greatest common divisor of two integers.
Let a and b be two integers . Then the greatest common divisor of a and b is
defined as the largest positive integer k such that a and b are both divisible by k.
Denoted by GCD(a,b). For instance, the gcd(2,4)=2
b) Describe at least three different ways to find the greatest common divisor of two
integers.
Method 1: Binary GCD
GCD(a,b)=GCD{min(a,b), |a-b|}
For instance, GCD(2,4)=GCD{min(2,4),|2-4|}=GCD(2,2)=2;
Method 2. The Euclidean algorithm
Given two natural numbers a and b, check if b is zero. If yes, a is the gcd. If not, repeat
the process using b and the remainder after integer division of a and b (written a modulus
b below).
function gcd(a, b)
if b = 0
return a
else
return gcd(b, a modulus b)
This can be rewritten iteratively as:
function gcd(a, b)
while b ≠ 0
var t := b
b := a modulus b
a := t
return a
Method 3: Given two integers a and b. First, we can write a and b in the standard
prime factorizations, namely,
a  p1 1 p2 2 ... pm m
r
r
r
and
b  q1 1 q2 2 ...qn n
l
l
l
where pi’s and qj’s are prime numbers. Then look at their factorizations and see
the common factors which are the GCD. In detailed, look at each of those pi and qj
such that pi=qj, then choose the smaller one.
For instance, to find GCD(135, 60)
135  33 * 5 and 60  2 2 * 3 * 5
So, the common factor is
3*5=15
So, GCD(135, 60)=15
c) Find the greatest common divisor of 1,234,567 and 7,654,321.
Use method 1.
GCD(1234567 ; 7654321)
=GCD(1234567 ; |7654321-1234567|)
=GCD(1234567 ; 6419754)
=GCD(1234567 ; |6419754-1234567|)
=GCD(1234567 ; 5185187)
=GCD(1234567 ; 3950620)
=GCD(1234567 ; 2716053)
=GCD(1234567 ; 1481486)
=GCD(1234567 ; 246919)
=GCD(246891; 246919)
=GCD(246891; 28)=GCD(8817*28+15; 28)
………………(1)
……………..
=GCD(15, 28)=1
………………………..(2)
Note: From (1) to (2), we have to repeat that process many times. Note that each time,
the bigger number decreased by 28, so it is good to write 246891=8817*28+15. So, after
8817 steps, it reduces to (2).
For (2), it is easy to see GCD(15,28)=1 since they don’t have a common factor other than
1.
d) Find the greatest common divisor of 2335577911 and 2937557313.
Use Method 3, we know that
GCD(2335577911 ; 2937557313)= 23355573
Note: just look at each term, for instance, in the first one, there is a factor 23
and the second one contains 25. We choose the smaller one 23 Keep doing this, we find
the largest common factor which is 23355573