• Title/Summary/Keyword: greatest common divisor

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An Investigation of Teaching Methods of Finding out the Greatest Common Divisor and the Least Common Multiple Focused on Their Meanings (최대공약수와 최소공배수를 구하는 과정에서 의미를 강조한 지도방안 탐색)

  • Pang, JeongSuk;Lee, YuJin
    • Journal of Elementary Mathematics Education in Korea
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    • v.22 no.3
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    • pp.283-308
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    • 2018
  • 'Divisor and multiple' is the topic included both in the elementary and in the secondary mathematics curriculum, but there has been lack of research on it. It has been reported that students have a difficulty in understanding the meaning of the greatest common divisor (GCD) and the least common multiple (LCM), while they can find out GCD and LCM. Against the lack of research on how to overcome this difficulty, this study designed teaching methods with a model for visualization to emphasize the meanings of divisor and multiple in finding out GCD and LCM, and implemented the methods in one fourth grade classroom. A questionnaire was developed to explore students' solution methods and interviews with focused students were implemented. In addition, fourth-grade students' thinking was compared and contrasted with fifth-grade students who studied divisor and multiple with the current textbook. The results of this study showed that the teaching methods with a specific model for visualization had a positive impact on students' conceptual understanding of the process to find out GCD and LCM. As such, this study provides instructional implications on how to foster the meanings of finding out GCD and LCM at the elementary school.

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ON THE GREATEST COMMON DIVISOR OF BINOMIAL COEFFICIENTS

  • Sunben Chiu;Pingzhi Yuan;Tao Zhou
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.4
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    • pp.863-872
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    • 2023
  • Let n ⩾ 2 be an integer, we denote the smallest integer b such that gcd {(nk) : b < k < n - b} > 1 as b(n). For any prime p, we denote the highest exponent α such that pα | n as vp(n). In this paper, we partially answer a question asked by Hong in 2016. For a composite number n and a prime number p with p | n, let n = ampm + r, 0 ⩽ r < pm, 0 < am < p. Then we have $$v_p\(\text{gcd}\{\(n\\k\)\;:\;b(n)1\}\)=\{\array{1,&&a_m=1\text{ and }r=b(n),\\0,&&\text{otherwise.}}$$

UNITARY ANALOGUES OF A GENERALIZED NUMBER-THEORETIC SUM

  • Traiwat Intarawong;Boonrod Yuttanan
    • Communications of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.355-364
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    • 2023
  • In this paper, we investigate the sums of the elements in the finite set $\{x^k:1{\leq}x{\leq}{\frac{n}{m}},\;gcd_u(x,n)=1\}$, where k, m and n are positive integers and gcdu(x, n) is the unitary greatest common divisor of x and n. Moreover, for some cases of k and m, we can give the explicit formulae for the sums involving some well-known arithmetic functions.

An Analysis of Teaching Divisor and Multiple in Elementary School Mathematics Textbooks (초등학교 수학 교과서에 나타난 약수와 배수지도 방법 분석)

  • Choi Ji Young;Kang Wan
    • Journal of Elementary Mathematics Education in Korea
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    • v.7 no.1
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    • pp.45-64
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    • 2003
  • This study analyzes divisor and multiple in elementary school mathematics textbooks published according to the first to the 7th curriculum, in a view point of the didactic transposition theory. In the first and second textbooks, the divisor and the multiple are taught in the chapter whose subject is on the calculations of the fractions. In the third and fourth textbooks, divisor and multiple became an independent chapter but instructed with the concept of set theory. In the fifth, the sixth, and the seventh textbooks, not only divisor multiple was educated as an independent chapter but also began to be instructed without any conjunction with set theory or a fractions. Especially, in the seventh textbook, the understanding through activities of students itself are strongly emphasized. The analysis on the each curriculum periods shows that the divisor and the multiple and the reduction of a fractions to the lowest terms and to a common denominator are treated at the same period. Learning activity elements are increase steadily as the textbooks and the mathematical systems are revised. The following conclusion can be deduced based on the textbook analysis and discussion for each curriculum periods. First, loaming instruction method also developed systematically with time. Second, teaching method of the divisor and multiple has been sophisticated during the 1st to 7th curriculum textbooks. And the variation of the teaching sequences of the divisor and multiple is identified. Third, we must present concrete models in real life and construct textbooks for students to abstract the concepts by themselves. Fourth, it is necessary to develop some didactics for students' contextualization and personalization of the greatest common divisor and least common multiple. Fifth, the 7th curriculum textbooks emphasize inquiries in real life which teaming activities by the student himself or herself.

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Hong Jung Ha's Number Theory (홍정하(洪正夏)의 수론(數論))

  • Hong, Sung-Sa;Hong, Young-Hee;Kim, Chang-Il
    • Journal for History of Mathematics
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    • v.24 no.4
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    • pp.1-6
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    • 2011
  • We investigate a method to find the least common multiples of numbers in the mathematics book GuIlJib(구일집(九一集), 1724) written by the greatest mathematician Hong Jung Ha(홍정하(洪正夏), 1684~?) in Chosun dynasty and then show his achievement on Number Theory. He first noticed that for the greatest common divisor d and the least common multiple l of two natural numbers a, b, l = $a\frac{b}{d}$ = $b\frac{a}{d}$ and $\frac{a}{d}$, $\frac{b}{d}$ are relatively prime and then obtained that for natural numbers $a_1,\;a_2,{\ldots},a_n$, their greatest common divisor D and least common multiple L, $\frac{ai}{D}$($1{\leq}i{\leq}n$) are relatively prime and there are relatively prime numbers $c_i(1{\leq}i{\leq}n)$ with L = $a_ic_i(1{\leq}i{\leq}n)$. The result is one of the most prominent mathematical results Number Theory in Chosun dynasty. The purpose of this paper is to show a process for Hong Jung Ha to capture and reveal a mathematical structure in the theory.

ABS ALGORITHMS FOR DIOPHANTINE LINEAR EQUATIONS AND INTEGER LP PROBLEMS

  • ZOU MEI FENG;XIA ZUN QUAN
    • Journal of applied mathematics & informatics
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    • v.17 no.1_2_3
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    • pp.93-107
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    • 2005
  • Based on the recently developed ABS algorithm for solving linear Diophantine equations, we present a special ABS algorithm for solving such equations which is effective in computation and storage, not requiring the computation of the greatest common divisor. A class of equations always solvable in integers is identified. Using this result, we discuss the ILP problem with upper and lower bounds on the variables.

A root finding algorithm of a polynomial over finite fields (유한체 위에서 다항식의 근에 관한 알고리즘)

  • 김창한
    • Journal of the Korea Institute of Information Security & Cryptology
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    • v.7 no.4
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    • pp.73-80
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    • 1997
  • 유한체 위에서 다항식의 근을 구하는 문제는 수학의 오래된 문제중 하나이고 최근들어 암호학과 관련하여 유한체 위서의 다항식 연산과 성질등이 쓰이고 있다. 유한체 위에서 다항식의 최대공약수(greatest common divisor) 를 구하는데 많은 시간이 소요 된다. Rabin의 알고리즘에서 주어진 다항식의 근들의 곱(F(x), $x^{q}$ -x)를 구하는 과정을 c F(p), $f_{c}$ (x)=(F(x), $T_{r}$ (x)-c), de$gf_{c}$ (x)>0인 $f_{c}$(x) s로 대체한 효율적인 알고리즘 제안과 Mathematica를 이용한 프로그램의 실행 결과를 제시한다.

Exploring Ways to Connect Conceptual Knowledge and Procedural Knowledge in Mathematical Modeling (수학적 모델링 수업에서 개념적 지식과 절차적 지식의 연결 방안 탐색)

  • Lee, Ye-jin;Choi, Mira;Kim, Yoonjung;Lim, Miin
    • Education of Primary School Mathematics
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    • v.26 no.4
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    • pp.349-368
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    • 2023
  • The purpose of this study is to explore ways for students to connect conceptual and procedural knowledge in mathematical modeling lessons. Accordingly, we selected the greatest common divisor among the learning contents in which elementary school students have difficulties connecting conceptual and procedural knowledge. A mathematical modeling lesson was designed and implemented to solve problems related to the greatest common divisor while connecting conceptual and procedural knowledge. As a result of the analysis, it was found that the mathematical modeling lesson had positive effects on students solving problems by connecting conceptual and procedural knowledge. In addition, through actual class application, a teaching and learning plan was derived to meaningfully connect conceptual and procedural knowledge in mathematical modeling lessons.

A Study of a Java Programming Plan for the Development of Mathematics Learning Materials of Middle School (중학교 수학 학습자료 개발을 위한 Java 프로그래밍 설계 연구)

  • 장진관
    • Journal of the Korean School Mathematics Society
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    • v.2 no.1
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    • pp.181-195
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    • 1999
  • This research is produced as a applet of learning materials, and is made with the internet languages HTML, Java, and NamoWebeditor. It contains "Greatest Common Divisor and Least Common Multiple", "Parallel translation of function of second order", "Pythagoras Theorem", which is the current middle school mathmatics textbook for third graders. The keynote of this research is that the students can study individually through logging into the internet on their own computers; the program is made using graphics and animation on order to develop the learners′ interest in mathematics. I hope that this research can supplement our currently insufficient internet educational data.

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