A simplified presentation for finding the LCM and the GCF

1974 ◽  
Vol 21 (5) ◽  
pp. 415-416
Author(s):  
Laurence Sherzer
Keyword(s):  
Common Factor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors ◽  
Positive Integers

Given the prime factors of two positive integers, the least common multiple (LCM) of these two numbers is the product of the union of these prime factors, and the greatest common factor (GCF) is the product of the intersection of these prime factors. If we could just state this fact to our students and be understood, our job of teaching them to find the LCM or the GCF of two numbers would be greatly simplified. Unfortunately, as in most teaching, simple verbal statements do not suffice.

On Nonsingular Power LCM Matrices

2006 ◽  
Vol 13 (04) ◽  
pp. 689-704 ◽  
Author(s):  
Shaofang Hong ◽  
K. P. Shum ◽  
Qi Sun
Keyword(s):  
Singular Number ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Closed Set ◽  
Prime Factors ◽  
The Matrix ◽  
Lcm Matrix ◽  
Positive Integers ◽  
Power Lcm Matrix

Let e ≥ 1 be an integer and S={x1,…,xn} a set of n distinct positive integers. The matrix ([xi, xj]e) having the power [xi, xj]e of the least common multiple of xi and xj as its (i, j)-entry is called the power least common multiple (LCM) matrix defined on S. The set S is called gcd-closed if (xi,xj) ∈ S for 1≤ i, j≤ n. Hong in 2004 showed that if the set S is gcd-closed such that every element of S has at most two distinct prime factors, then the power LCM matrix on S is nonsingular. In this paper, we use Hong's method developed in his previous papers to consider the next case. We prove that if every element of an arbitrary gcd-closed set S is of the form pqr, or p2qr, or p3qr, where p, q and r are distinct primes, then except for the case e=1 and 270, 520 ∈ S, the power LCM matrix on S is nonsingular. We also show that if S is a gcd-closed set satisfying xi< 180 for all 1≤ i≤ n, then the power LCM matrix on S is nonsingular. This proves that 180 is the least primitive singular number. For the lcm-closed case, we establish similar results.

Sharing Teaching Ideas

1988 ◽  
Vol 81 (8) ◽  
pp. 648-652
Author(s):  
Paul A. Klein
Keyword(s):  
Common Factor ◽  
Common Denominator ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors ◽  
Instructional Sequences ◽  
Teaching Ideas

Few arithmetic skills have eluded as many students as those used in reducing fractions or in obtaining the least common denominator of two fractions. In the more popular instructional sequences the general concepts of greatest common factor (GCF) and least common multiple (LCM) are developed first Usually, these concepts involve the decomposition of the numbers into prime factors as shown in example 1.

Geometric Figures Make the LCM Obvious

1987 ◽  
Vol 34 (7) ◽  
pp. 17-18
Author(s):  
Flo McEnery Edwards
Keyword(s):  
Common Factor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Geometric Figures ◽  
Prime Factors

The greatest common factor (GCF) is easily found by comparing the prime factors of a set of numbers and choosing the prime factors they have in common. The least common multiple (LCM) is often more difficult for students to find. The prime factors of the LCM are not necessarily common to a set of numbers, but their product i a common multiple of those numbers. According to Troutman and Lichtenberg (1982, 173), “the least multiple that is common to some set of numbers will have all the factors of those numbers and no other factors.” How, then, is it possible to find these prime factors just by inspection? One way is to write the prime factors of a set of numbers in a list and use geometric figures to find the prime factors of the LCM.

A Variation on the Algorithm for GCD and LCM

1982 ◽  
Vol 30 (3) ◽  
pp. 46
Author(s):  
Verna M. Adams
Keyword(s):  
Prime Numbers ◽  
Prime Divisors ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors

An algorithm sometimes presented for finding the least common multiple (LCM) of two numbers uses tbe technique of simultaneously finding the prime factors of the numbers. This technique is shown in figure 1. Both numbers are checked for divisibility by 2, then by 3, by 5, and so on. If the divisor does not divide one of the numbers, the number is written on the next line as shown in steps 4 and 5. This process continues until all numbers to the left and on the bottom are prime numbers, or it can be continued, as shown in figure 1, until the numbers across the bottom are all ones. The least common multiple is the product of all of the prime divisors. Thus, LCM (80, 72) = 24 · 32 · 5.

MORE ON A CERTAIN ARITHMETICAL DETERMINANT

2017 ◽  
Vol 97 (1) ◽  
pp. 15-25 ◽  
Author(s):  
ZONGBING LIN ◽  
SIAO HONG
Keyword(s):  
Linear Algebra ◽  
Acta Math ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Mobius Function ◽  
Dirichlet Convolution ◽  
Positive Integers

Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$, where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$. In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

scholarly journals The Implementation of APIQ Creative Mathematics Game Method in the Subject Matter of Greatest Common Factor and Least Common Multiple in Elementary School

2017 ◽  
Author(s):  
Ansari Saleh Ahmar ◽  
Abdul Rahman ◽  
Andi Nurani Mangkawani Arifin ◽  
Dewi Satria Ahmar ◽  
M. Agus ◽  
...  
Keyword(s):  
Elementary School ◽  
Subject Matter ◽  
Common Factor ◽  
Traditional Learning ◽  
Learning Method ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Causal Factors ◽  
Learning Mathematics ◽  
The Subject

One of causal factors for uninterested feeling of the students in learning mathematics is a monotonous learning method, like in traditional learning method. One of the ways for motivating students to learn mathematics is by implementing APIQ (Aritmetika Plus Intelegensi Quantum) creative mathematics game method. The purposes of this research are (1) to describe students’ responses toward the implementation of APIQ creative mathematics game method on the subject matter of Greatest Common Factor (GCF) and Least Common Multiple (LCM) and (2) to find out whether by implementing this method, the student’s learning completeness will improve or not. Based on the results of this research, it is shown that the responses of the students toward the implementation of APIQ creative mathematics game method in the subject matters of GCF and LCM were good. It is seen in the percentage of the responses were between 76-100%. (2) The implementation of APIQ creative mathematics game method on the subject matters of GCF and LCM improved the students’ learning.

On the divisibility among power GCD and power LCM matrices on gcd-closed sets

2021 ◽  
pp. 1-12
Author(s):  
Guangyan Zhu
Keyword(s):  
Greatest Common Divisor ◽  
Text Entry ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Closed Set ◽  
Closed Sets ◽  
Matrix Formula ◽  
Power Matrix ◽  
Positive Integers

Let [Formula: see text] and [Formula: see text] be positive integers and let [Formula: see text] be a set of [Formula: see text] distinct positive integers. For [Formula: see text], one defines [Formula: see text]. We denote by [Formula: see text] (respectively, [Formula: see text]) the [Formula: see text] matrix having the [Formula: see text]th power of the greatest common divisor (respectively, the least common multiple) of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry. In this paper, we show that for arbitrary positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], the [Formula: see text]th power matrices [Formula: see text] and [Formula: see text] are both divisible by the [Formula: see text]th power matrix [Formula: see text] if [Formula: see text] is a gcd-closed set (i.e. [Formula: see text] for all integers [Formula: see text] and [Formula: see text] with [Formula: see text]) such that [Formula: see text]. This confirms two conjectures of Shaofang Hong proposed in 2008.

Greatest Common Factor and Least Common Multiple

1984 ◽  
Vol 31 (8) ◽  
pp. 43-44
Author(s):  
Charles E. Lamb ◽  
Lyndal R. Hutcherson
Keyword(s):  
Common Factor ◽  
Computational Algorithms ◽  
Common Multiple ◽  
Least Common Multiple

The greatest common factor and the least common multiple are two concepts that are important in their own right as well as crucial to the development of other topics in mathematics. Unfortunately, these two topics can be difficult for students to understand even apart from the process of computing numerical values for them. This article discusses some strategies for avoiding misconceptions of these ideas and reviews some computational algorithms for them.

Teacher to Teacher: Finding Common Ground

1999 ◽  
Vol 5 (4) ◽  
pp. 236
Author(s):  
Elizabeth H. Bradley
Keyword(s):  
Common Ground ◽  
Common Factor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Visual Aid

IF YOUR STUDENTS CONFUSE GREATEST common factor (GCF) with least common multiple (LCM), try this approach, which employs a diagram as a visual aid and can be used to reduce fractions.

scholarly journals Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

2014 ◽  
Vol 57 (3) ◽  
pp. 551-561 ◽  
Author(s):  
Daniel M. Kane ◽  
Scott Duke Kominers
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Finite Arithmetic ◽  
Positive Integers

AbstractFor relatively prime positive integers u0 and r, we consider the least common multiple Ln := lcm(u0, u1..., un) of the finite arithmetic progression . We derive new lower bounds on Ln that improve upon those obtained previously when either u0 or n is large. When r is prime, our best bound is sharp up to a factor of n + 1 for u0 properly chosen, and is also nearly sharp as n → ∞.

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