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

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## Details

### 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*

### Abstract

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 *Z _{n}* 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

*Z*for various values of n. Further, we present an algorithm for determining all these pairs in

_{n}*Z*.

_{n}### Keywords

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

### 2010 Mathematics Subject Classification

- 97K20
- 97F60
- 11A07

### References

- Rosen, K. H. (2019). Elementary Number Theory and Its Applications, 6th Edition. Pearson New International Edition.
- 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.
- Beachy, J. A., & Blair, W. D. (2006). Abstract Algebra, 3rd Edition. Waveland Press Inc.
- Shan, Z., Wang, E. T. H. (1999). Mutual multiplies in
*Z*. Mathematics Magazine, 72 (2), 143–145._{n} - Buck, W. K. (2004). Cyclic Rings. Master Thesis, Eastern Illinois University.
- Sajana, S., & Bharathi, D. (2019). Number theoretic properties of the commutative ring
*Z*. Int. J. Res. Ind. Eng. 8 (1), 77–88._{n}

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## Cite this paper

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