MUHAMMAD YUSRAN BASRI
MATH ICP B 2012
1211441023
SUMMARY
EUCLIDEAN ALGORITHM
A useful algorithm for finding the greatest comon divisor of two positive integers 𝑚 and 𝑛 is
the Euclidean Algorithm. It consists of repeated application of the division algorithm:
𝑚 = 𝑛𝑞1 + 𝑟1 , 1 ≤ 𝑟1 ≤ 𝑛,
𝑛 = 𝑟1 𝑞2 + 𝑟2 , 1 ≤ 𝑟2 < 𝑟1 ,
...
𝑟𝑘−2 = 𝑟𝑘−1 𝑞𝑘 + 𝑟𝑘 , 1 ≤ 𝑟𝑘 < 𝑟𝑘−1 ,
𝑟𝑘−1 = 𝑟𝑘 𝑞𝑘+1 + 𝑟𝑘+1 , 𝑟𝑘+1 = 0,
The last non zero remainder,𝑟𝑘 is the greatest common divisor of m and n.
a. We will find GCD(90, 78) with no sigh factors.
Step 1: Apply the division algorithm on 90 and 78
90 = 1 . 78 + 12
0 ≤r< 78
Step 2: Apply the division algorithm on 78 and 12
78 = 6 . 12 + 6
0 ≤r< 12
Step 3: Apply the division algorithm on 12 and 6
12 = 2 . 6
0 ≤r< 6
We can obtain that the GCD(90, 78) = 6
1. Lemma:If a = qb + r, then GCD(a, b) = GCD(b, r).
Proof:
Suppose GCD(a, b) = d. And GCD(b, r) = d
(a) Because GCD(a, b) = d, then d|a and d| b.
From here we obtain d | a and d|qb. so d|(a – qb) or d|r thus, d|b dan d|r (1)
(b) Suppose c|b dan c|r. From here we obtain c|qb. So that c|(qb + r) or c|a.
Since GCD(a, b) = d, then for c|a and c|b we will obtain c ≤ d. So, if
c|b and c|r then c ≤ d (2)
From (1) and (2) concluded that the GCD (b, r) = d = GCD (a, b)
b. We can search GCD(a, b) without registering the factors.
Step 1: Apply the division algorithm at a and b
a = q1 . b + r1
0 ≤r1<b
Step 2: Apply the division algorithm at b dan r1
b = q2 .r1 + r2
0 ≤r2<r1
Step 3: Apply the division algorithm at r1 dan r2
r1 = q3 .r2 + r3
0 ≤r3<r2
Step n+1: Apply the division algorithm at rn dan rn-1
rn-1 = q3 .rn
0 ≤rn+1<rn
GCD(a, b) = GCD(b, r1) = . . . = GCD(rn-1 , rn ) = GCD(rn , 0) = rn
Illustration
Find GCD(1769, 2378). Then find the integers number x and y so that GCD(1769, 2378) =
1769x + 2378y
Answer
29 = 551 – 9.58
= 551 – 9.(609 – 1.551)
= 10.551 – 9.609
= 10.(1769 – 2.609) – 9.609
= 10.1769 – 29.609
= 10.1769 – 29.(2378 – 1.1769)
= 1769(39) + 2378(–29)
so, the value x = 39 and y = –29
2378 = 1 . 1769 + 609
1769 = 2 . 609 + 551
609 = 1 . 551 + 58
551 = 9 . 58 + 29
58 = 2 . 29
So, GCD (1769, 2378) = 29