Classical pairs in Zn

Tekuri Chalapathi, Shaik Sajana, and Dasari Bharathi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 59—69
DOI: 10.7546/nntdm.2020.26.1.59-69
Download full paper: PDF, 716 Kb


Authors and affiliations

Tekuri Chalapathi
Department of Mathematics, Sree Vidyanikethan Eng. College
Tirupati, Andhra Pradesh., India

Shaik Sajana
Department of Mathematics, P.R. Govt. College (A)
Kakinada, Andhra Pradesh., India

Dasari Bharathi
Department of Mathematics, S. V. University
Tirupati, Andhra Pradesh., India


The interplay between algebraic structures and their elements have been the most
famous and productive area of the algebraic theory of numbers. Generally, the greatest
common divisor and least common multiple of any two positive integers are dependably
non-zero elements. In this paper, we introduce a new pair of elements, called classical pair in the ring Zn whose least common multiple is zero and concentrate the properties of these pairs. We establish a formula for determining the number of classical pairs in Zn for various values of n. Further, we present an algorithm for determining all these pairs in Zn.


  • Greatest common divisor
  • Least common multiple
  • Euler-totient function
  • Classical pairs

2010 Mathematics Subject Classification

  • 97K20
  • 97F60
  • 11A07


  1. Rosen, K. H. (2019). Elementary Number Theory and Its Applications, 6th Edition. Pearson New International Edition.
  2. Chalapathi, T., & Kiran Kumar, R. V. M. S. S. (2016). Graph structures of Euler totient numbers. Daffodil International Journal of Science and Technology. 11 (2), 19–29.
  3. Beachy, J. A., & Blair, W. D. (2006). Abstract Algebra, 3rd Edition. Waveland Press Inc.
  4. Shan, Z., Wang, E. T. H. (1999). Mutual multiplies in Zn. Mathematics Magazine, 72 (2), 143–145.
  5. Buck, W. K. (2004). Cyclic Rings. Master Thesis, Eastern Illinois University.
  6. Sajana, S., & Bharathi, D. (2019). Number theoretic properties of the commutative ring Zn. Int. J. Res. Ind. Eng. 8 (1), 77–88.

Related papers

Cite this paper

Chalapathi, T., Sajana, S., & Bharathi, D. (2020). Classical pairs in Zn. Notes on Number Theory and Discrete Mathematics, 26(1), 59-69, doi: 10.7546/nntdm.2020.26.1.59-69.

Comments are closed.