A Tutorial on New Methods and Algorithms for Solving LCM and GCD
M. SABRIGIRIRAJ
Department of Information Technology,
Hindusthan College of Engineering and Technology, Coimbatore,
INDIA
Abstract: - This paper presents a new tutorial on the innovative methods and algorithms for solving the
commonly used mathematical problems namely Least Common Multiple (LCM) and Greatest Common Divisor
(GCD). Starting with the basic concepts, the tutorial takes the readers through various new techniques with
step-by-step example implementations for both LCM and GCD. Appropriate pseudocodes for the proposed
techniques based on iterative and Euclidean approaches are presented towards effective problem-solving. The
pseudocodes presented are focused on using functions and customized to solve problems with larger size inputs
and thereby facilitate scalability. A performance comparison study is also done to evaluate the efficiency of the
proposed methods. This tutorial article provides a solid base for readers aspiring to master the art of problem-
solving and programming with a systematic approach.
Key-Words: - LCM, GCD, Programming, Algorithms, Pseudocode, Problem-Solving.
Received: October 15, 2023. Revised: May 15, 2024. Accepted: May 17, 2024. Published: June 20, 2024.
1 Introduction
LCM (Least Common Multiple) and GCD (Greatest
Common Divisor) are two basic mathematical
operations that find extensive applications in
Engineering, Science, Management, and
Technology. In particular, LCM and GCD are
employed during the optimization of resources in
several critical applications ranging from electrical
motor design, cloud computing, cyber-physical
systems, and radar engineering to the Industrial
Internet of things. The problem of LCM and GCD
finds extensive attention from researchers in various
perspectives ranging from focusing on improved
teaching and learning outcomes to applying in a
variety of domains focusing on real-world
applications.
Researchers have given enormous attention to
studying and exploring the concepts of LCM and
GCD over the past years. Research on the
verification of sentences involving LCM is explored
in [1]. A detailed examination of the necessity of
computational thinking in primary mathematics
education, along with ways to integrate it into the
teaching and learning process, is provided in [2].
[3], presents an interactive computer-based learning
approach for LCM and GCD using animations. [4],
focuses on the development of an educational game
media based on an Android application for effective
learning of LCM and GCD. The importance of
developing a mathematics textbook using realistic
teaching methods to motivate elementary school
students is discussed in [5]. Problem-based learning
of LCM and GCD is shown to be more effective and
attractive for school students, [6]. A report on the
Indonesian Realistic Mathematics Education
approach, [7] and the safari numbered model, [8],
demonstrates a positive effect on students’ learning
outcomes regarding GCD and LCM in
primaryschool. A study [9], [10], presents the
broader picture of computers drastically changing
mathematics learning and enhancing the potential of
computational thinking in formal school settings.
The primary building blocks of teaching LCM and
GCD are disseminated in [11]. [12], successfully
demonstrates a gaming technique for teaching LCM
and GCD to improve learning outcomes.
Research [13], delves into a method to process
repetition in musical composition using LCM,
highlighting the close relationship between
mathematics and music. [14], exhibits finding LCM
and GCD using DNA computing. A realistic
mathematical approach, [15], [16], [17], [18] is
employed to teach LCM and GCD, increasing
students' reasoning abilities. Computational thinking
increases while solving LCM and GCD with
Scratch, as demonstrated in [19]. [20], introduces a
digital creative magic e-book for learning
mathematical concepts, including LCM and GCD,
designed and adopted for school students. New
teaching methods for LCM and GCD are researched
with a focus on their meanings, [21]. Proactive
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interference is noted in [22], when students use
LCM strategies to solve GCD problems.
The effectiveness of a mathematics teacher’s
actions prompting various thoughts is examined in
[23], while teaching LCM and GCD. The
underlying issues for misconceptions among
students during the computation of LCM and GCD
are analyzed in [24], [25]. Common errors made by
students while solving LCM and GCD, along with
other learning obstacles and troubleshooting tips,
are analyzed in [26], [27], [28], [29]. [30], employs
three different visual models to teach LCM and
GCD, conducting a comparative study. A set of real
problems involving LCM and GCD in practice is
demonstrated in [31]. The application of LCM and
GCD for computing distractor suites in developing
choices for a multiple-choice question is
demonstrated in [32]. LCM and GCD are used as
sample mathematical concepts to assess the
performance of the ChatGPT-4 code interpreter,
[33]. They also find a place in the performance
evaluation of a multispace model, [34], along with
network security and cryptography techniques, [35],
[36].
Although a vast literature is available
concentrating on teaching-learning methods for
LCM and GCD using books, eBooks, interactive
mode learning techniques, gaming methods,
problem-solving techniques, magic books,
animation software, in addition to ways of tracing
misconceptions and troubleshooting methods, still
there is a scope to expand it from a programming
point of view. Expanding the teaching and learning
methods of LCM and GCD from a programming
perspective increases the computational thinking,
logical thinking, project management, and problem
solving capabilities of the readers. To address this
research gap, the authors have explored and
introduced innovative techniques from a
programming standpoint. In this paper, the concepts
of LCM and GCD have been dealt with through
simple straightforward examples, and new methods
have been suggested from a programming
perspective. Additionally, a performance
comparison study is conducted among the proposed
techniques.
The rest of this article is prepared as follows:
Sections 2 through 3 illustrate LCM and GCD
through simple examples and new methods and
algorithms are proposed with appropriate
pseudocodes. Performance comparison between the
proposed techniques is carried out in section 4.
Section 5 concludes the paper and points out future
research directions.
2 Least Common Multiple (LCM)
Least Common Multiple (LCM) is a mathematical
technique to find the smallest common multiple for
a set of given numbers. It is very trivial to note that
the lowest common multiple may lie anywhere
between the biggest input (among the given set of
numbers) and the product of all the given numbers.
Example 1. Find the LCM of a) 4 and 6 b) 4 and 5
c) 5 and 7 d) 2 and 6. Write a pseudocode for the
same.
Solution:
a) LCM of 4 and 6 needs to be a number
between 6 and 24. Then, consider the series
6, 7, 8, 9, …, 24. In this series, 12 is the
smallest number that divides both inputs 4
and 6 without leaving any remainder. So,
LCM is 12.
b) 20
c) 35
d) 6
Algorithm LCM_1 (a, b)
{
Max = maximum (a, b)
while ((Max % a ≠ 0) ‘OR’ (Max % b ≠ 0)
Max +=1
return (Max)
}
The above pseudocode Algorithm LCM_1 (a, b)
may be easily extended for any number of inputs
and it is very trivial.
An alternative technique to find LCM is to
begin by comparing one’s multiple of all the given
inputs. When the one’s multiple of the inputs is not
equal, the multiplying integer values on the smaller
sides are incremented from their current value by 1
and the comparison is again done. This process is
repeated again and again until all sides become
equal. This is illustrated below in Example 2.
Example 2: Find the LCM of 4 and 6. Write a
pseudocode for the same.
Solution:
4 *1 < 6*1
4 *2 > 6*1
4*2 < 6*2
4*3 = 6 *2
12 = 12
Algorithm LCM_2 (a, b)
{
Input a, b
Initialize i = j =1
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while (a*i ≠ b*j)
if (a*i > b*j)
j ++
else
i ++
return (a*i)
}
Example 3: Find the LCM of 2, 3, 4. Write a
pseudocode for the same.
2 * 1 < 3 * 1 < 4 * 1
2 * 2 < 3 * 2 > 4 * 1
2 * 3 = 3 * 2 < 4 * 2
2 * 4 < 3 * 3 > 4 * 2
2 * 5 > 3 * 3 < 4 * 3
2 * 6 = 3 * 4 = 4 * 3
12 = 12 = 12
Algorithm LCM_3 (a, b, c)
{
Input a, b, c,
Initialize i = j = k = 1
while (a*i ≠ b*j ≠ c*k)
if (a*i > b*j) ‘AND’ (a*i > c*k)
j ++, k ++
else if ( b*j > a*i) ‘AND’ (b*j > c*k)
i ++, k ++
else if (c*k > a*i) ‘AND’ (c*k > b*j)
i ++, j ++
else if (a*i > b*j) ‘AND’ (a*i = = c*k)
j ++
else if (b*j > a*i) ‘AND’ (b*j = = c*k)
i ++
else
k ++
return (a*i)
}
The above pseudocodes, Algorithm LCM_2 (a,
b) and Algorithm LCM_3 (a, b, c) may be extended
for any number of inputs, but the number of ‘else if’
conditions will exponentially increase (and so code
length increases) as the input size increases.
However, the concept of ‘function call’ may be
employed to find the LCM of a bigger size input set
because LCM (a, b, c) = LCM (LCM (a, b), c).
Here, the code length remains the same irrespective
of the input size. The pseudocode for a set of N
numbers stored as an array or list is given below.
Algorithm LCM_N (A [1…N])
{
K = A [1]
for (i = 2; i ≤ N; i ++)
K= LCM_1 (K, A[i])
return (K)
}
It can be easily deduced that the time
complexity for finding the LCM of N given
numbers is O(N*K). Also, the time complexity of
finding the LCM of two input numbers is O(K) and
K denotes the approximate number of comparisons
involved.
3 Greatest Common Divisor (GCD)
GCD (also known as the highest common factor or
HCF) is the biggest positive integer that divides the
given set of positive integers exactly without
leaving any remainder. It is very trivial to note that
any common divisor for the given set of input must
lie anywhere between the smallest input (among the
given set of input numbers) and unity.
Example 4. Find the GCD of a) 4 and 6 b) 4 and 5
c) 5 and 7 d) 2 and 6. Write a pseudocode for the
same.
Solution:
a) GCD of 4 and 6 need to be a number
between 4 and 1. Then, consider the series
4,3,2,1. In this series, 2 is the biggest
number that divides both inputs 4 and 6
without leaving any remainder. So, GCD is
2.
b) 1
c) 1
d) 2
Algorithm GCD_1 (a, b)
{
Min = minimum (a, b)
while ((a % Min ≠ 0) ‘OR’ (b % Min ≠ 0))
Min - = 1
return (Min)
}
The above pseudocode Algorithm GCD_1 (a, b)
may be easily extended for any number of inputs
and it is very trivial.
An alternative technique to solve the 2-input
GCD problem is repeated subtraction, [15]. In the
first step, the bigger input number needs to be
subtracted the smaller input number and the result to
be saved. For each of the subsequent steps, the first
biggest number among the three (two input numbers
and one saved result of the previous step) needs to
be discarded and the second biggest number has to
subtract the smallest number and the result to be
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saved. The above step needs to be repeated again
and again till the subtracted result reaches zero. It is
important to note that GCD (m, 0) = m. This is
illustrated in the following Example 5.
Example 5: Find the GCD of 12 and 3 by repeated
subtraction method. Write a pseudocode for the
same.
GCD (12, 3) = GCD (12-3, 3)
GCD (9, 3) = GCD (9-3, 3)
GCD (6, 3) = GCD (6-3, 3)
GCD (3, 3) = GCD (3-3, 3)
GCD (0, 3) = 3
Algorithm GCD_2 (a, b)
{
Input a, b
while (a ≠ b)
if (a > b)
a = a – b
else
b = b - a
return (a)
}
This technique can be easily extended for 3
input problems and is illustrated in Example 6.
Example 6: Find the GCD of 12, 36, and 3 by
repeated subtraction method. Write a pseudocode
for the same.
GCD (12, 36, 3) = GCD (12, 36-12, 3)
GCD (12, 24, 3) = GCD (12, 24-12, 3)
GCD (12, 12, 3) = GCD (12-12, 12, 3)
GCD (0, 12, 3) = GCD (0, 12-3, 3)
GCD (0, 9, 3) = GCD (0, 9-3, 3)
GCD (0, 6, 3) = GCD (0, 6-3, 3)
GCD (0, 3, 3) = GCD (0, 3-3, 3)
GCD (0, 0, 3) = 3
Algorithm GCD_3 (a, b, c)
{
Input a, b, c
while ( (a = b ≠ 0) ‘OR’ (b = c ≠ 0) ‘OR’ (a = c ≠ 0) )
if (a > b) and (a > c) and (b > c)
a = a – b
elseif (a > b) and (a > c) and (b <= c)
a = a – c
elseif (b > a) and (b > c) and (a > c)
b = b - a
elseif (b > a) and (b > c) and (a <= c)
b = b - c
elseif (c > a) and (c > b) and (a > b)
c = c – a
elseif (c > a) and (c > b) and (a <= b)
c = c – b
elseif (a == b) and (a < c)
a = a – b
elseif (b == c) and (b < a)
b = b – c
else
c = c – a
return (maximum (a, b, c))
}
The above pseudocodes, Algorithm GCD_2 (a,
b) and Algorithm GCD_3 (a, b, c) may be extended
for any number of inputs, but the number of ‘else if’
conditions will exponentially increase (and so code
length increases) as the input size increases.
However, the concept of ‘function call’ may be
employed to find the GCD of a bigger size input set
because GCD (a, b, c) = GCD (GCD (a, b), c).
Here, the code length remains the same irrespective
of the input size.
The pseudocode for a set of N numbers stored
as an array or list is given below.
Algorithm GCD_N (A [1…N])
{
K = A [1]
for (i = 2; i ≤ N; i ++)
K= GCD_1 (K, A[i])
return (K)
}
Another alternative technique to solve the 2-input
GCD problem is Euclid’s method [16]. The basic
concept of the Euclidean algorithm is given as
follows:
GCD (m, n) = GCD (n, m mod n)
GCD (m, 0) = m
Example 7: Find the GCD of 8 and 12 by Euclid’s
method. Write a pseudocode for the same.
Solution:
GCD (8,12) = GCD (12, 8)
GCD (12, 8) = GCD (8, 4)
GCD (8, 4) = GCD (4, 0)
GCD (4, 0) = 0
Hence, GCD (8, 12) = 4
Algorithm GCD_3 (a, b)
{
while (b ≠ 0)
(a, b) = (b, a % b)
return (a)
}
It can be easily deduced that the time complexity for
finding the GCD of N given numbers is O(N*K).
Also, the time complexity of finding the GCD of
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two input numbers is O(K) and K denotes the
approximate number of comparisons involved.
4 Results and Discussion
A detailed programming study was carried out to
evaluate the performance of the proposed techniques
for LCM and GCD in terms of their execution time.
The evaluation of the proposed techniques was
carried out using an HP Laptop configured with a
core i3 processor with 8GB RAM and 512 MB SSD.
Coding was done using Python programming
language. The inputs for finding LCM and GCD
were randomly generated using the built-in
random.randint() function in python. Input size was
varied between 2 to 25 for the evaluation study. To
evaluate the proposed LCM techniques, function
calls were made to LCM_1(a, b) and LCM_2(a, b).
With the same inputs, function calls were also made
to GCD_1(a, b), GCD_2(a, b), and GCD_3(a, b) to
evaluate the proposed GCD techniques. Table 1 and
Table 2 show the execution time for computing
LCM and GCD respectively with the proposed
techniques. From Table 1, it can be noted that
LCM_2(a, b) executed faster compared to
LCM_1(a, b) because LCM_2(a, b) employs fewer
iterations than LCM_1(a, b). From Table 2, it can be
noted that GCD_3(a, b) executed faster when
compared to GCD_1(a, b) and GCD_2(a, b) because
GCD_3(a, b) employs fewer iterations than
GCD_1(a, b) and GCD_2(a, b). Another noteworthy
observation is that program execution time does not
increase linearly with input size. It is likely due to
the random nature of inputs and also depends on the
availability of common factors. In some cases, even
if the input size is less, due to the absence of
common factors, both LCM and GCD consume
more execution time when compared to a higher
number of inputs with common factors.
Table 1. Execution time with the proposed LCM
techniques
Input
Size
Execution time in seconds
LCM_1(a, b)
2
1.4543*10-5
3
6.6757*10-6
4
1.9311*10-5
5
1.4305*10-5
6
1.6450*10-5
7
2.3126*10-5
8
5.3644*10-5
9
1.5258*10-5
10
2.6941*10-5
15
3.3140*10-5
20
2.5749*10-5
25
8.7976*10-5
Table 2. Execution time with the proposed GCD
techniques
Input
Size
Execution time in microseconds
GCD_1(a, b)
GCD_2(a, b)
GCD_3(a, b)
2
1.0967*10-5
8.5830*10-6
4.2915*10-6
3
1.5497*10-5
8.3446*10-6
2.6226*10-6
4
1.6689*10-5
1.2159*10-5
7.8678*10-6
5
2.0265*10-5
7.3909*10-6
4.7683*10-6
6
1.4305*10-5
1.2636*10-5
9.0599*10-6
7
2.0265*10-5
1.0251*10-5
7.6293*10-6
8
1.3113*10-5
1.3351*10-5
6.6757*10-6
9
1.1682*10-5
1.1205*10-5
9.0599*10-6
10
1.6927*10-5
1.1682*10-5
7.1525*10-6
15
4.2676*10-5
2.5272*10-5
1.7881*10-5
20
2.8848*10-5
1.8835*10-5
1.3828*10-5
25
3.6954*10-5
2.3365*10-5
2.3126*10-5
5 Conclusion
This tutorial paper revisited LCM and GCD through
simple examples and proposed new ways to solve
them, along with appropriate pseudocodes. A
comparative study was also conducted to evaluate
the supremacy of the proposed methods. There
exists a huge scope to extend this paper in a
multitude of domains. Firstly, the similar study may
be conducted to explore other common
mathematical operations from a programming
perspective. Secondly, LCM and GCD operations
may be implemented in VLSI technology and a
performance evaluation study to be done. Thirdly,
the implementation of LCM and GCD using
quantum computing circuits is to be investigated.
Fourthly, implementing LCM and GCD using
optical computing techniques may be explored.
Finally, applications of artificial intelligence and
computational techniques with LCM and GCD are
an exciting area for exploration.
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