• Journal of the Chungcheong Mathematical Society
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• v.30 no.4
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• pp.449-457
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• 2017

In this paper we study several generalizations of greatest common divisor of GCD domain with always the greatest common divisor.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1215-1219
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• 2008

We find a necessary and sufficient condition that the prime factors of $A^m+1$ and $A^n+1$ coincide for odd positive integers $n>m{\geq}1$. Moreover, we also find a necessary and sufficient condition that the set of all prime factors of $A^m+1$ is a subset of those of $A^n+1$ for $n>m{\geq}1$.

• 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.

• 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.