The Left Greatest Common Divisor and the Left Least Common Multiple for All Solutions of the Matrix Equation BX = A Over a Commutative Domain of Elementary Divisors

2021 ◽  
Author(s):  
V. P. Shchedryk
Keyword(s):  
Matrix Equation ◽  
Greatest Common Divisor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Elementary Divisors ◽  
All Solutions ◽  
The Matrix ◽  
Commutative Domain

scholarly journals Relations Between Ranks of Matrix Polynomials

2021 ◽  
Vol 24 (1) ◽  
pp. 35-41
Author(s):  
Vasile Pop ◽  
Keyword(s):  
Greatest Common Divisor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Matrix Polynomials ◽  
The Matrix

We show that the sum of the ranks of two matrix polynomials is the same as the sum of the rank of the matrix obtained by applying the greatest common divisor of the polynomials, with the rank of the matrix obtained by applying the least common multiple of the polynomials. Many applications, for older or more recent problems, of this result are obtained.

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.

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.

scholarly journals The Effect of Guided Inquiry Learning Model to the Metacognitive Ability of Primary School Students

2020 ◽  
Vol 8 (1) ◽  
pp. 37
Author(s):  
Intan Dwi Hastuti ◽  
Yuni Mariyati ◽  
S. Sutarto ◽  
Chairun Nasirin
Keyword(s):  
Primary School ◽  
Inquiry Learning ◽  
Learning Model ◽  
Greatest Common Divisor ◽  
Guided Inquiry ◽  
Primary School Students ◽  
School Students ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Metacognitive Ability

This study aimed at analyzing the effect of guided inquiry learning to the metacognitive ability of primary school students on the material of Least Common Multiple (KPK) and Greatest Common Divisor (FPB). The type of study was a mixed-method using quantitative and qualitative methods. There were 55 students of 4th grade used as the subjects of study. Two learning models were compared, namely guided inquiry learning model and conventional learning model. The students’ metacognitive ability was measured by means of problem-solving test on the material of Least Common Multiple (KPK) and Greatest Common Divisor (FPB). The quantitative analysis data used descriptive and inferential statistical tests. According to the results of data analysis, it was discovered that the t-test of sig (2-tailed) from the independent samples t-test of post-test was 0,00 (p = <0,05); this indicated that there was a significant difference on it. This showed that there was a difference of students’ metacognitive ability for both classes in solving the problems of Least Common Multiple (KPK) and Greatest Common Divisor (FPB) after the guided inquiry learning was implemented. Consequently, it can be concluded that there is a significant effect on the implementation of guided inquiry learning model to improve the students’ metacognitive ability in solving the material problems of Least Common Multiple (KPK) and Greatest Common Divisor (FPB).

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.

scholarly journals Developing a Math Textbook using realistic Mathematics Education Approach to increase elementary students’ learning motivation

2021 ◽  
Vol 9 (2) ◽  
Author(s):  
Jesi Alexander Alim ◽  
Neni Hermita ◽  
Melvi Lesmana Alim ◽  
Tommy Tanu Wijaya ◽  
Jerito Pereira
Keyword(s):  
Mathematics Education ◽  
Research And Development ◽  
Elementary Students ◽  
Learning Motivation ◽  
Realistic Mathematics Education ◽  
Greatest Common Divisor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Realistic Mathematics

This study aims to develop an appropriate and practical math textbook in the unit of the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) using Realistic Mathematics Education (RME) in order to increase elementary students’ learning motivation. This is a Research and Development (RnD) type of study with the Plomp model. A mathematician and a teacher assessed the validity of the textbook. The practicality of the textbook was assessed by two teachers and 15 students using questionnaires. The students' motivation was assessed by the students using questionnaires as well. The results showed that the textbook was appropriate with an average of 83.32%, the respondent results from the students’ views were practical with an average of 82.33% and very practical with an average of 87.6 from the teachers’ view. This study also found that the textbook increased the students’ learning motivation by 6.45%.

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