Reflection Borders for Patchwork Quilts

1986 ◽  
Vol 79 (2) ◽  
pp. 138-143
Author(s):  
Duane DeTemple
Keyword(s):  
Problem Solving ◽  
Basic Program ◽  
Greatest Common Divisor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Problem Solving Skills ◽  
General Proof ◽  
Special Cases ◽  
Fertile Area

The question posed in this article provides an unexpectedly fertile area in which students can test their problem-solving skills. The problem is easily stated and understood, and special cases can be quickly generated as a source of conjectures and possible counterexamples. The solution is elusive but elegantly simple, and students may well guess the answer long before they produce a general proof. The solution involves the concepts of least common multiple and greatest common divisor, even though the problem itself is essentially a geometric one. An appendix by James H. Jordan lists an Apple BASIC program that permits students to experiment and gather data by means of microcomputer graphics. Figures 1 and 2 are printouts from this program.

Another Look at Least Common Multiple and Greatest Common Factor

1978 ◽  
Vol 25 (6) ◽  
pp. 52-53
Author(s):  
Loren L. Henry
Keyword(s):  
Middle School ◽  
Problem Solving ◽  
Common Factor ◽  
Middle School Mathematics ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factorization ◽  
Problem Solving Skills ◽  
Whole Numbers ◽  
Main Disadvantage

Most contemporary middle school mathematics programs use the notion of prime factorization to obtain the least common multiple and the greatest common factor of a pair of natural or whole numbers. Success in determining the least common multiple or greatest common factor for a pair of numbers then depends on the ability to obtain the prime factorization for any given number. The main disadvantage of this approach arises when either of the pair of numbers is quite large. For example, determining the least common multiple of 2464 and 7469 by prime factorization involves knowledge of divisibility tests for primes and of the primes themselves. The purpose of this paper is to examine another method for obtaining the least common multiple and greatest common factor of a pair of numbers without using prime factorization. These methods provide opportunities for students to make and test conjectures about the possible generalizations of the results to more than two numbers. Such conjecturing should aid students in developing problem-solving skills.

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).

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%.

Star Patterns

1978 ◽  
Vol 25 (4) ◽  
pp. 12-14
Author(s):  
Albert B. Bennett
Keyword(s):  
Mathematical Activity ◽  
Greatest Common Divisor ◽  
Common Multiple ◽  
Least Common Multiple

Making star patterns such as those pictured in figures 1 and 2 is a common mathematical activity for producing colorful and artistic patterns. Often, however, such patterns are constructed with very little use of mathematics beyond counting. The purpose of this article is to show how the concepts of greatest common divisor and least common multiple are related to star patterns.

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