2017 International Conference on Advanced Education and Management Science (AEMS 2017)
ISBN: 978-1-60595-438-7
On the Very Useful Euclidean Algorithm and Greatest Common Factor
Li-jiang ZENG
Research Centre of Zunyi Normal College, Zunyi 563099, GuiZhou, China
Keywords: Euclidean algorithm, Greatest common factor, Natural numbers, Subtractions.
Abstract. For going to do good science research, it must use mathematical tools. However, almost
all branch of mathematics, cannot leave the greatest common factor and Euclidean algorithm.
Euclidean algorithm is the most basic and the most important role. If there is no Euclidean
algorithm, almost every branch of mathematics is not in-depth research. In this paper, we introduce
the Euclidean algorithm, as well as the method of Euclidean algorithm is used to determine the
greatest common factor of natural numbers.
Introduction
Almost all natural science cannot leave the mathematics[1-7], someone once said: science no
mathematical tools is the science of imperfect. However, mathematics initially is roughly divided
into two teams, algebra and geometry. Almost all more detailed branch of algebra cannot leave the
greatest common factor[8-11]. If without a greatest common factor one couldn't in-depth study[12-14].
To resolve the greatest common factor problem depends on the Euclidean algorithm. With the
development of modern science, not only is the direction of algebra, even the direction of the
geometry problems often need the greatest common factor and Euclidean algorithm[15-17]. This
paper is to introduce the Euclidean algorithm and the method using Euclidean algorithm determines
greatest common factor.
Some Preparation
If natural numbers a and b have a common divisor d, then
a = a ' d and b = b ' d for some natural numbers a' and b'. From this it follows that d divides a –
b, because
a − b = a ' d − b ' d = (a '− b ')d
In other words, a common divisor of a and b is also a divisor of a-b.
Euclid used this fact to find the greatest common divisor[1-7] gcd(a,b), by “repeatedly subtracting
the smaller number from the larger”. More precisely, his algorithm goes as follows.
The Euclidean Algorithm and Greatest Common Factor
Suppose that a>b and let: a1 = a , b1 = b . Then for each pair ( ai , bi ) , we .form the pair
( ai +1 , bi +1 ) , where
ai +1 = max(bi , ai − bi ) ,
bi +1 = min(bi , ai − bi ) .
Since this process produces smaller and smaller natural number, it must halt (by "descent"). we
eventually get ak = b k , in which case we conclude that gcd(a, b) = ak = b k .
The reason this algorithm works is that
gcd( a1 , b1 ) = gcd(a2 , b2 ) = = gcd(ak , bk )
…
since any common divisor of the pair ( a1 , b1 ) is also a divisor of the pairs ( a2 , b2 ) , ( a3 , b3 ) , ,
( ak , bk ) produced by the successive subtractions.
56
The Example for Illustration of Algorithm
In order to more clearly illustrate methods, we combined with the specific examples to illustrate the
Euclidean algorithm.
Example 1. a=34, b=19
The algorithm gives the following pairs:
( a1 , b1 ) = (34, 19)
( a2 , b2 ) = (19, 34 − 19) = (19, 15)
( a3 , b3 ) = (15, 19 − 15) = (15, 4)
( a4 , b4 ) = (15 − 4, 4) = (11, 4)
( a5 , b5 ) = (11 − 4, 4) = (7, 4)
( a6 , b6 ) = (4, 7 − 4) = (4, 3)
( a7 , b7 ) = (3, 4 − 3) = (3, 1)
( a8 , b8 ) = (3 − 1, 1) = (2, 1)
(a9 , b9 ) = (2 − 1, 1) = (1, 1)
and therefore gcd(34, 19) = gcd(l,1) = l.
Integer pairs a, b such that gcd(a, b)=1 are said to be relatively prime. Thus the Euclidean
algorithm gives a simple means of deciding whether integers are relatively prime. In the next
section, we see that the algorithm (in a slightly modified form) is also highly efficient: it gives
gcd(a,b) in a number of steps comparable with the total number of digits in a and b. It is harder to
recognize whether a single integer n is prime: the obvious methods require a number of steps
comparable with the size of n, which is exponentially larger-around 2k if k is the number of binary
digits of n.
The Gcd by Division with Remainder
Euclid's form of the gcd algorithm is usually speeded up by doing division with remainder instead
of repeated subtraction. Given a pair ( ai , bi ) with ai > bi , the next pair is produced by the rule
，b
= remainder when ai is divided by bi . This is more efficient when ai is many
times as large as bi , in which case many subtractions are replaced by one division. However, the
algorithm is essentially the same—division of natural number is just repeated subtraction—so it is
still true that
ai +1 = bi
i +1
gcd( a1 , b1 ) = gcd(a2 , b2 ) = The only difference is that halting now occurs when bk divides ak , in which case we conclude
that gcd( a, b) = gcd(ak , bk ) = bk .
Example 2. a = 34 , b = 19 again
The algorithm with division gives the following pairs:
( a1 , b1 ) = (34, 19)
( a2 , b2 ) = (19, 34 − 19) = (19, 15)
( a3 , b3 ) = (15, 19 − 15) = (15, 4)
( a4 , b4 ) = (4, 15 − 3 × 4) = (4, 3)
( a5 , b5 ) = (3, 4 − 3) = (3, 1)
Hence gcd(34, 19) = 1 because l divides 3.
57
Conclusions
In this form of the algorithm it is easy to see that the number of division is comparable with the total
number of digits in a and b. In fact, if a and b are written in binary, then each division. Reduces the
total number of digits by at least one. If a has more digits than b this is clear: the new pair is b
together with a remainder on division by b that has no more digits than b. If a and b have the same
number of digits then, since both a and b necessarily begin with the digit l, the remainder is simply
a-b, and it has fewer digits than b.
The division form of the Euclidean algorithm is not only more efficient; it also has wider
applicability. For example, in Z[i] we can divide 17 by 4+i (exactly) and get the quotient 4-i, but it
is meaningless to subtract 4+i from 17“4-i times". Thus division in Z[i] is not generally the result of
repeated subtraction. Any Euclidean algorithm in Z[i] necessarily uses division with remainder.
Acknowledgement
Author introduction: Lijiang Zeng (1962 -), male, born in Chishui of Guizhou Province, Professor
of Zunyi Normal College, major research field: mathematics and applied mathematics, research
direction: algebra and its application, number theory and its application, function theory and
application. Have existed search results: CPCI-S(ISTP), CPCI-SSH(ISSHP), and EI 17 articles
published, 26 English papers published.
Reference
[1] Clarke, George, Xu, Lixin Colin, and Zou, Heng-fu. Finance and Income Inequality: Test of
Alternative Theories[C]. World Bank Policy Research Working Paper, 2003, 2984-2988.
[2] Yushuang Fan, Weidong Gao, Qinghai Zhong. On the Erdos–Ginzburg–Ziv constant of finite
abelian groups of high rank[J]. Journal of Number Theory . 2011 (10) , 99-102.
[3] Blank, B, Gilson, R. Venture Capital and the Structure of Capital Markets: Blanks Versus Stock
Markets[J]. The Journal of Finance. 1998(3), 89-92
[4] Xiangneng Zeng, Pingzhi Yuan. Weighted Davenport’s constant and the weighted EGZ
Theorem[J]. Discrete Mathematics . 2011 (17) , 74-79.
[5] Sukumar Das Adhikari,Yong-Gao Chen. Davenport constant with weights and some related
questions, II[J]. Journal of Combinatorial Theory, Series A . 2007 (1), 104-109.
[6] Olson, J.E. A combinatorial problem on finite abelian groups (I, II) [J]. Journal of Number
Theory. 1969(2), 56-81.
[7] P. Baginski, S. T. Chapman, K. McDonald, L. Pudwell. On cross numbers of minimal zero
sequences in certain cyclic groups[J]. Ars Combinatoria. 2004(2), 56-61.
[8] Michael F. Barnsley.
1986 (1), 89-99.
Fractal functions and interpolation[J]. Constructive Approximation .
[9] Win M, Pinto P, Shepp L .A Mathematical Theory of Net-work Interference and Its
Applications[C]. Proceedings of the IEEE. 2009, 82-87.
[10] Musibau Ibrahim, Ramakrishnan Mukundan. Multi-fractal Techniques for Emphysema
Classification in Lung Tissue Images[A]. Proceedings of 2014 3rd International Conference on
Environment, Chemistry and Biology[C]. 2014, 1121-1126.
[11] Yi-ru Huang, Jian Huang, Jian-sheng Yang College of Sciences, Shanghai University
Shanghai, China, 200444Yan-ping Liu School of Statistics, Renmin University of China, Beijing,
China, 100872. Fractal—Research on the Complex Carry System and its Decimal Set[A].
58
Proceedings of 2010 International Workshop on Chaos-Fractals Theories and Applications[C].
2010, 1021-1026.
[12] D. K. Tsang, S. Oyadiji, A. Leung. Dynamic Analysis of a Penny-shaped Crack by the
Fractal-like Finite Element Method[A]. Proceedings of the 5th International Conference on
Vibration Engineering[C]. 2002, 103-107.
[13] Kang Lin, Li Chengmao. The Application of Fractal Theory in Graphics Design[A].
Manufacturing Systems and Industry Application ICMEAT 2011 MSIA [C]. 2011,1203-1207
（
）
[14] Domke G, Laskar R. The bondage and reinforcement number of γ f for some graphs[J].
Discrete Mathematics. 1997, 101-104.
[15] Fink J, Jacobson M, Kinch L, et al. The bondage number of a graph[J]. Discrete Mathematics .
1990, 91-94.
[16] Liu X, Li Y. The Incidence Coloring Number of Complete k-Partite Graph[J]. Journal of China
University of Mining & Technology. 2001(01), 93-97.
[17] HU G. Catalan Number and Enumeration of Maximal Outerplanar Graphs[J]. Tsinghua
Science and Technology. 2000(01),121-124.
59