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Journal articles on the topic 'Common divisor'

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Published: 4 June 2021
Last updated: 15 February 2022

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1

Beslin, Scott, and Steve Ligh. "Greatest common divisor matrices." Linear Algebra and its Applications 118 (June 1989): 69–76. http://dx.doi.org/10.1016/0024-3795(89)90572-7.

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2

Belenkiy, Ari, and Raimundas Vidunas. "A Greatest Common Divisor Algorithm." International Journal of Algebra and Computation 08, no. 05 (October 1998): 617–23. http://dx.doi.org/10.1142/s0218196798000296.

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Algorithms of computation of the Greatest Common Divisor (GCD) of two integers play a principal role in all computational systems dealing with rational arithmetic. The simplest one (Euclidean) is not the best for large numbers (see D. E. Knuth's book "The Art of Computer Programming" for details). One improvement was suggested by D. H. Lehmer in 1938 who noticed that it is possible to run the Euclidean algorithm with a few leading digits of large numbers and, with some care, still obtain the correct answer. In the 70's G. E. Collins pointed out that Lehmer's algorithm simultaneously analyzed two similar sequences of numbers and hence did twice as much work as necessary. Collins found a way to work with only one sequence of numbers together with a verification of a certain inequality. The proof of the validity of this inequality is, perhaps, too complicated. We present a similar but softer inequality and give a short and simple proof thereof.
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3

Koryukin, A. N., A. M. Sebeldin, and A. L. Sylla. "Rings with the greatest common divisor." Journal of Mathematical Sciences 183, no. 3 (May 3, 2012): 319–22. http://dx.doi.org/10.1007/s10958-012-0817-0.

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4

Lindqvist, Peter, and Kristian Seip. "Note on some greatest common divisor matrices." Acta Arithmetica 84, no. 2 (1998): 149–54. http://dx.doi.org/10.4064/aa-84-2-149-154.

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5

Zhukov, Kirill Dmitrievich. "Approximate common divisor problem and lattice sieving." Matematicheskie Voprosy Kriptografii [Mathematical Aspects of Cryptography] 9, no. 2 (2018): 87–98. http://dx.doi.org/10.4213/mvk257.

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6

Galbraith, Steven D., Shishay W. Gebregiyorgis, and Sean Murphy. "Algorithms for the approximate common divisor problem." LMS Journal of Computation and Mathematics 19, A (2016): 58–72. http://dx.doi.org/10.1112/s1461157016000218.

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The security of several homomorphic encryption schemes depends on the hardness of variants of the approximate common divisor (ACD) problem. We survey and compare a number of lattice-based algorithms for the ACD problem, with particular attention to some very recently proposed variants of the ACD problem. One of our main goals is to compare the multivariate polynomial approach with other methods. We find that the multivariate polynomial approach is not better than the orthogonal lattice algorithm for practical cryptanalysis.We also briefly discuss a sample-amplification technique for ACD samples and a pre-processing algorithm similar to the Blum–Kalai–Wasserman algorithm for learning parity with noise. The details of this work are given in the full version of the paper.
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7

Pollack, Paul. "On the greatest common divisor of a number and its sum of divisors." Michigan Mathematical Journal 60, no. 1 (April 2011): 199–214. http://dx.doi.org/10.1307/mmj/1301586311.

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8

Oka, Satomi. "On the common divisor of discriminants of integers." Tsukuba Journal of Mathematics 26, no. 1 (June 2002): 69–78. http://dx.doi.org/10.21099/tkbjm/1496164382.

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9

Olson, Melfried. "Activities: A Geometric Look at Greatest Common Divisor." Mathematics Teacher 84, no. 3 (March 1991): 202–8. http://dx.doi.org/10.5951/mt.84.3.0202.

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10

Heyman, Randell, and Igor E. Shparlinski. "On the greatest common divisor of shifted sets." Journal of Number Theory 154 (September 2015): 63–73. http://dx.doi.org/10.1016/j.jnt.2015.02.012.

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11

Peters, James V. "Millikan's oil droplet experiment and common divisor probabilities." International Journal of Mathematical Education in Science and Technology 22, no. 6 (November 1991): 873–76. http://dx.doi.org/10.1080/0020739910220602.

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12

Kim, Ji-Hyun. "Spurious correlation between ratios with a common divisor." Statistics & Probability Letters 44, no. 4 (October 1999): 383–86. http://dx.doi.org/10.1016/s0167-7152(99)00030-9.

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13

Ghaffarzadeh, Mehdi, Mohsen Ghasemi, and Mark L. Lewis. "Finite Groups Whose Common-Divisor Graph is Regular." Proceedings of the Edinburgh Mathematical Society 61, no. 2 (February 15, 2018): 329–41. http://dx.doi.org/10.1017/s0013091517000141.

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AbstractLet G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).
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14

Carl McTague. "On the Greatest Common Divisor of Binomial Coefficients." American Mathematical Monthly 124, no. 4 (2017): 353. http://dx.doi.org/10.4169/amer.math.monthly.124.4.353.

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15

Ramkumar, R. S., and K. Nagarajan. "Greatest common divisor degree estrada index of graphs." Journal of Discrete Mathematical Sciences and Cryptography 22, no. 5 (July 4, 2019): 825–33. http://dx.doi.org/10.1080/09720529.2019.1685235.

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16

Jakimczuk, Rafael. "The greatest common divisor of k positive integers." International Mathematical Forum 13, no. 5 (2018): 215–23. http://dx.doi.org/10.12988/imf.2018.8212.

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17

Hong, Siao. "The greatest common divisor of certain binomial coefficients." Comptes Rendus Mathematique 354, no. 8 (August 2016): 756–61. http://dx.doi.org/10.1016/j.crma.2016.06.001.

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18

Bhat, B. V. Rajarama. "On greatest common divisor matrices and their applications." Linear Algebra and its Applications 158 (November 1991): 77–97. http://dx.doi.org/10.1016/0024-3795(91)90051-w.

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19

Szegedy, M. "The solution of graham’s greatest common divisor problem." Combinatorica 6, no. 1 (March 1986): 67–71. http://dx.doi.org/10.1007/bf02579410.

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20

Bosek, Bartłomiej, Michał Dębski, Jarosław Grytczuk, Joanna Sokół, Małgorzata Śleszyńska-Nowak, and Wiktor Żelazny. "Graph coloring and Graham’s greatest common divisor problem." Discrete Mathematics 341, no. 3 (March 2018): 781–85. http://dx.doi.org/10.1016/j.disc.2017.11.006.

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21

Isaacs, I. M., and C. E. Praeger. "Permutation Group Subdegrees and the Common Divisor Graph." Journal of Algebra 159, no. 1 (August 1993): 158–75. http://dx.doi.org/10.1006/jabr.1993.1152.

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22

Kaplan, Gil. "On Groups Admitting a Disconnected Common Divisor Graph." Journal of Algebra 193, no. 2 (July 1997): 616–28. http://dx.doi.org/10.1006/jabr.1996.6992.

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23

BERGMAN, GEORGE M. "ON COMMON DIVISORS OF MULTINOMIAL COEFFICIENTS." Bulletin of the Australian Mathematical Society 83, no. 1 (October 13, 2010): 138–57. http://dx.doi.org/10.1017/s0004972710001723.

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AbstractErdős and Szekeres [‘Some number theoretic problems on binomial coefficients’, Aust. Math. Soc. Gaz.5 (1978), 97–99] showed that for any four positive integers satisfying m1+m2=n1+n2, the two binomial coefficients (m1+m2)!/m1!m2! and (n1+n2)!/n1!n2! have a common divisor greater than 1. The analogous statement for k-element families of k-nomial coefficients (k>1) was conjectured in 1997 by David Wasserman.Erdős and Szekeres remark that if m1,m2,n1,n2 as above are all greater than 1, there is probably a lower bound on the common divisor in question which goes to infinity as a function of m1 +m2 . Such a bound is obtained in Section 2.The remainder of this paper is devoted to proving results that narrow the class of possible counterexamples to Wasserman’s conjecture.
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24

GOMEZ, DOMINGO, JAIME GUTIERREZ, ÁLVAR IBEAS, and DAVID SEVILLA. "COMMON FACTORS OF RESULTANTS MODULO p." Bulletin of the Australian Mathematical Society 79, no. 2 (March 13, 2009): 299–302. http://dx.doi.org/10.1017/s0004972708001275.

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AbstractWe show that the multiplicity of a prime p as a factor of the resultant of two polynomials with integer coefficients is at least the degree of their greatest common divisor modulo p. This answers an open question by Konyagin and Shparlinski.
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25

Krätzel, Ekkehard, Werner Georg Nowak, and László Tóth. "On certain arithmetic functions involving the greatest common divisor." Central European Journal of Mathematics 10, no. 2 (January 7, 2012): 761–74. http://dx.doi.org/10.2478/s11533-011-0144-6.

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26

Altarawneh, Haroon. "A Comparison of Several Greatest Common Divisor 'GCD' Algorithms." International Journal of Computer Applications 26, no. 5 (July 31, 2011): 24–31. http://dx.doi.org/10.5120/3099-4253.

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27

HONG, SHAOFANG, and RAPHAEL LOEWY. "ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF GREATEST COMMON DIVISOR MATRICES." Glasgow Mathematical Journal 46, no. 3 (September 2004): 551–69. http://dx.doi.org/10.1017/s0017089504001995.

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28

Sarkar, Santanu, and Subhamoy Maitra. "Approximate Integer Common Divisor Problem Relates to Implicit Factorization." IEEE Transactions on Information Theory 57, no. 6 (June 2011): 4002–13. http://dx.doi.org/10.1109/tit.2011.2137270.

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29

Usevich, Konstantin, and Ivan Markovsky. "Variable projection methods for approximate (greatest) common divisor computations." Theoretical Computer Science 681 (June 2017): 176–98. http://dx.doi.org/10.1016/j.tcs.2017.03.028.

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30

Tembhurne, Jitendra V., and Shailesh R. Sathe. "New Modified Euclidean and Binary Greatest Common Divisor Algorithm." IETE Journal of Research 62, no. 6 (September 7, 2016): 852–58. http://dx.doi.org/10.1080/03772063.2016.1216809.

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31

Mansour, Yishay, Baruch Schieber, and Prasoon Tiwari. "A lower bound for integer greatest common divisor computations." Journal of the ACM (JACM) 38, no. 2 (April 1991): 453–71. http://dx.doi.org/10.1145/103516.103522.

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32

Luca, Florian. "On the greatest common divisor of two Cullen numbers." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 73, no. 1 (December 2003): 253–70. http://dx.doi.org/10.1007/bf02941281.

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33

Eslahchi, Ch, and A. M. Rahimi. "Thek-Zero-Divisor Hypergraph of a Commutative Ring." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–15. http://dx.doi.org/10.1155/2007/50875.

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The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and thek-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. LetRbe a commutative ring andkan integer strictly larger than2. Ak-uniform hypergraphHk(R)with the vertex setZ(R,k), the set of allk-zero-divisors inR, is associated toR, where eachk-subset ofZ(R,k)that satisfies thek-zero-divisor condition is an edge inHk(R). It is shown that ifRhas two prime idealsP1andP2with zero their only common point, thenHk(R)is a bipartite (2-colorable) hypergraph with partition setsP1−Z′andP2−Z′, whereZ′is the set of all zero divisors ofRwhich are notk-zero-divisors inR. IfRhas a nonzero nilpotent element, then a lower bound for the clique number ofH3(R)is found. Also, we have shown thatH3(R)is connected with diameter at most 4 wheneverx2≠0for all3-zero-divisorsxofR. Finally, it is shown that for any finite nonlocal ringR, the hypergraphH3(R)is complete if and only ifRis isomorphic toZ2×Z2×Z2.
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34

Ray, Prasanta Kumar. "Greatest common divisors of shifted balancing numbers." Boletim da Sociedade Paranaense de Matemática 35, no. 3 (October 25, 2017): 273. http://dx.doi.org/10.5269/bspm.v35i3.26093.

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It is well known that the successive balancing numbers are relatively prime. Let for all integers a, sn(a) denotes the greatest common divisor of the shifted balancing numbers of the form sn(a) = gcd(Bn 􀀀 a; Bn+1 􀀀 6a).In this study, we will show that fsn(1)g is unbounded whereas fsn(a)g isbounded for a 6= 1.
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35

Halikias, G., G. Galanis, N. Karcanias, and E. Milonidis. "Nearest common root of polynomials, approximate greatest common divisor and the structured singular value." IMA Journal of Mathematical Control and Information 30, no. 4 (November 11, 2012): 423–42. http://dx.doi.org/10.1093/imamci/dns032.

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36

Haukkanen, Pentti, and Varanasi Sitaramaiah. "Bi-unitary multiperfect numbers, V." Notes on Number Theory and Discrete Mathematics 27, no. 2 (June 2021): 20–40. http://dx.doi.org/10.7546/nntdm.2021.27.2.20-40.

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A divisor $d$ of a positive integer $n$ is called a unitary divisor if $\gcd(d, n/d)=1;$ and $d$ is called a bi-unitary divisor of $n$ if the greatest common unitary divisor of $d$ and $n/d$ is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let $\sig^{**}(n)$ denote the sum of the bi-unitary divisors of $n$. A positive integer $n$ is called a bi-unitary multiperfect number if $\sig^{**}(n)=kn$ for some $k\geq 3$. For $k=3$ we obtain the bi-unitary triperfect numbers. Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is part V in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we determined all bi-unitary triperfect numbers of the form $n=2^{a}u$, where $1\leq a \leq 6$ and $u$ is odd. In parts IV(a-b) we solved partly the case $a=7$. In this paper we fix the case $a=8$. In fact, we show that $n=57657600=2^{8}.3^{2}.5^{2}.7.11.13$ is the only bi-unitary triperfect number of the present type.
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37

Ahlswede, Rudolf, and Levon Khachatrian. "Sets of integers and quasi-integers with pairwise common divisor." Acta Arithmetica 74, no. 2 (1996): 141–53. http://dx.doi.org/10.4064/aa-74-2-141-153.

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38

Fatouros, Stavros, Nicos Karcanias, Dimitrios Christou, and Pericles Papadopoulos. "Approximate Greatest Common Divisor of Many Polynomials and Pseudo-Spectrum." IFAC Proceedings Volumes 46, no. 2 (2013): 623–28. http://dx.doi.org/10.3182/20130204-3-fr-2033.00113.

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39

Moschovakis, Yiannis N. "On primitive recursive algorithms and the greatest common divisor function." Theoretical Computer Science 301, no. 1-3 (May 2003): 1–30. http://dx.doi.org/10.1016/s0304-3975(02)00487-5.

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40

Duzhin, S. V., and S. V. Chmutov. "Gaydar's formula for the greatest common divisor of several polynomials." Russian Mathematical Surveys 48, no. 2 (April 30, 1993): 171–72. http://dx.doi.org/10.1070/rm1993v048n02abeh001018.

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41

JOHNSON, D. S., A. C. PUGH, and G. E. HAYTON. "Symbolic calculation of greatest common divisor of 2D polynomial matrices." IMA Journal of Mathematical Control and Information 12, no. 1 (1995): 5–15. http://dx.doi.org/10.1093/imamci/12.1.5.

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42

Smadi, Mahmoud A., Ibrahim Marouf, Mohammad M. Asad, and Qasem Abu Al Haija. "Design alternatives of Euclidian greatest common divisor with enhanced architecture." International Journal of Computer Aided Engineering and Technology 13, no. 4 (2020): 425. http://dx.doi.org/10.1504/ijcaet.2020.10029304.

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43

Haija, Qasem Abu Al, Mohammad M. Asad, Ibrahim Marouf, and Mahmoud A. Smadi. "Design alternatives of Euclidian greatest common divisor with enhanced architecture." International Journal of Computer Aided Engineering and Technology 13, no. 4 (2020): 425. http://dx.doi.org/10.1504/ijcaet.2020.110480.

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44

Cheon, Jung Hee, Wonhee Cho, Minki Hhan, Jiseung Kim, and Changmin Lee. "Algorithms for CRT-variant of Approximate Greatest Common Divisor Problem." Journal of Mathematical Cryptology 14, no. 1 (October 20, 2020): 397–413. http://dx.doi.org/10.1515/jmc-2019-0031.

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AbstractThe approximate greatest common divisor problem (ACD) and its variants have been used to construct many cryptographic primitives. In particular, the variants of the ACD problem based on Chinese remainder theorem (CRT) are being used in the constructions of a batch fully homomorphic encryption to encrypt multiple messages in one ciphertext. Despite the utility of the CRT-variant scheme, the algorithms that secures its security foundation have not been probed well enough.In this paper, we propose two algorithms and the results of experiments in which the proposed algorithms were used to solve the variant problem. Both algorithms take the same time complexity $\begin{array}{} \displaystyle 2^{\tilde{O}(\frac{\gamma}{(\eta-\rho)^2})} \end{array}$ up to a polynomial factor to solve the variant problem for the bit size of samples γ, secret primes η, and error bound ρ. Our algorithm gives the first parameter condition related to η and γ size. From the results of the experiments, it has been proved that the proposed algorithms work well both in theoretical and experimental terms.
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45

Schinzel, A. "On the greatest common divisor of two univariate polynomials, II." Acta Arithmetica 98, no. 1 (2001): 95–106. http://dx.doi.org/10.4064/aa98-1-6.

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46

Christou, Dimitrios, Marilena Mitrouli, and Dimitrios Triantafyllou. "Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials." Special Matrices 5, no. 1 (October 26, 2017): 202–24. http://dx.doi.org/10.1515/spma-2017-0015.

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Abstract This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical effciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of theGCDof sets of several polynomials with real coeffcients is provided.Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented.
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47

Vijayakumar, P., S. Bose, and A. Kannan. "Centralized key distribution protocol using the greatest common divisor method." Computers & Mathematics with Applications 65, no. 9 (May 2013): 1360–68. http://dx.doi.org/10.1016/j.camwa.2012.01.038.

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48

Joris, H., C. Oestreicher, and J. Steinig. "The greatest common divisor of certain sets of binomial coefficients." Journal of Number Theory 21, no. 1 (August 1985): 101–19. http://dx.doi.org/10.1016/0022-314x(85)90013-7.

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49

Hastuti, Intan Dwi, Yuni Mariyati, S. Sutarto, and Chairun Nasirin. "The Effect of Guided Inquiry Learning Model to the Metacognitive Ability of Primary School Students." Prisma Sains : Jurnal Pengkajian Ilmu dan Pembelajaran Matematika dan IPA IKIP Mataram 8, no. 1 (July 13, 2020): 37. http://dx.doi.org/10.33394/j-ps.v8i1.2615.

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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).
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50

Haukkanen, Pentti, and Varanasi Sitaramaiah. "Bi-unitary multiperfect numbers, IV(b)." Notes on Number Theory and Discrete Mathematics 27, no. 1 (March 2021): 45–69. http://dx.doi.org/10.7546/nntdm.2021.27.1.45-69.

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A divisor d of a positive integer n is called a unitary divisor if \gcd(d, n/d)=1; and d is called a bi-unitary divisor of n if the greatest common unitary divisor of d and n/d is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let \sig^{**}(n) denote the sum of the bi-unitary divisors of n. A positive integer n is called a bi-unitary multiperfect number if \sig^{**}(n)=kn for some k\geq 3. For k=3 we obtain the bi-unitary triperfect numbers. Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is part IV(b) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we considered bi-unitary triperfect numbers of the form n=2^{a}u, where 1\leq a \leq 6 and u is odd. In part IV(a) we solved partly the case a=7. We proved that if n is a bi-unitary triperfect number of the form n=2^{7}.5^{b}.17^{c}.v, where (v, 2.5.17)=1, then b\geq 2. We then solved completely the case b=2. In the present paper we give some partial results concerning the case b\ge 3 under the assumption 3\nmid n.
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